MANU-MEN TAL 


COMPUTATION 


WOODFORD D. ANDERSON 


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MATHEMATICS LISRARY 


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FIGURE /, 


MANU-MENTAL 
COMPUTATION 


Bx 


Wooprorp D. ANDERSON, A.M., PH.D. 
Commercial Department 
Girls’ ‘Technical High School 
New York City 


FORMERLY PROFESSOR IN 
MISSOURI WESLEYAN COLLEGE 
MORNINGSIDE COLLEGE 

; AND 
UNIVERSITY OF SOUTH DAKOTA 


DRAWINGS BY 


CLEMENT B. DAVIS 


New YorK, 1904 


\ 


Copyright, .1 
¢ 
y Ween NDERSON. 


Entered at Stationers’ Hall, London, 1904. 


The Greenwich Press 
New York 


BA H\A AS 
A. 2 MATHEMATICS LIBRARY 


Ba ar SE ase. patie 
Co DY Mother, 

Who Patiently Driller JHle in the 
first Principles of S#athematics, 
Chis Book Fs Affectionatelp 

| Dedicatey, 


250607 


INTRODUCTION. 


In this day of advanced methods every progressive 
teacher will welcome any device that will save time and 
increase accuracy. 

The author’s experience as teacher in one ungraded 
country school, one graded school, three high schools, 
two colleges and one State university, has convinced him 
that most students of arithmetic (possibly in most mathe- 
matics) are slaves to the pencil or chalk, and are unable 
to reason independently or to remember their results 
longer than it requires to write them. He believes the 
student should be taught “mental” or ‘‘intellectual”’ 
arithmetic until he is able to reason clearly, compute 
accurately and remember results until the desired con- 
clusion is obtained. 

On several occasions the author has suggested to men of 
recognized mathematical ability that the use of pencil and 
chalk should be discouraged among pupils, and has 
invariably been met by the reply that students cannot 
add large numbers without aids, nor spare time enough to 
learn the multiplication table to 100 times 100, and that 
few students could remember it if it were taught. 

The author set to work to discover some method by 
which the student could add large numbers without the 
use of the pencil or chalk, and found the “joints” and 
“balls” of the fingers admirably adaptable for this proc- 
ess. After a brief trial he discovered that subtraction, 
multiplication, and division could be readily performed 
by the same method. 


6 INTRODUCTION. 


This system gradually developed itself until, by the use 
of the fingers, the average student can— 


1. Read all numbers below one quadrillion as accu- 
rately as he can to one thousand, and save at least half the 
time required to learn it. 


2. Write all numbers below one quadrillion as accu- 
rately as he can to one thousand, and save more than half 
of the time needed to learn it. 


3. Write decimals from the decimal point without 
having to enumerate to place the point. 


4, Read decimals without pausing to read the denom- 
inator. 


5. Add any numbers to the sum of one million. (By 
continued use of this method the student can soon learn 
to add and subtract numbers of two, three, or four figures 
mentally.) ; 


6. Subtract where neither number exceeds one million. 


7. Multiply any numbers whose product does not exceed 
one million. 


8. Divide one million, or less, by any number below it. 


9. Solve most practical problems in arithmetic without 
the use of pencil or chalk. 


10. Cultivate the memory and develop marvelous 
mathematical skill. 

The teacher should be sure that he understands the 
principles thoroughly, and can perform the operations 
readily. He should be careful to see that the student 
does the same. 

In presenting this volume to the public, the author 
does not claim to present a complete arithmetic; nor does 
he desire that it shall be substituted for a textbook; but 
he hopes that its use, in connection with textbooks now 
in use, will revolutionize the teaching of arithmetic and 


INTRODUCTION. 7 


greatly increase the independence and efficiency of the 
student and business man. 

The problems given are intended to show what can be 
done by the Manu-Mental method, and to suggest the 
kind of problems necessary for the student to gain pro- 
ficiency. Where one is given in this book, a dozen or more 
similar ones may be selected from the textbook. 

While the author claims originality in the Manu-Mental 
method, he desires to express his obligation to his teachers 
who helped him to think independently ; toSuperintendent 
Wm. E. Chancellor, of Bloomfield, N. J., for advice 
concerning name and publication, and to the students of 
the Bloomfield High School (N. J.) and Girls’ Technical 
High School (New York City) for clerical work and test- 
ing the Manu-Mental method. 

The author realizes that it would be almost impossible 
to present a new method of computation without errors; he 
therefore invites corrections, criticisms, and suggestions, 
and will gladly answer any questions concerning the work. 


THE AUTHOR. 
New York, July 4, 1904. 


CONTENTS. 
aM ER ETSEN Sete cre Co eee te ee Sens ork eee o 
ASME STEM PTSIOIN TS PC cee Cac Ae pe Sen yi oe od hms 14 
STK HATING rik tan ie. ete es Cee Se eee 15 
MeeeaLY FAL) LG ER a ee hoa ee Sag wade ee ead wee 16 
6 TMi gS, et CO 8 a gery aR AL REESE RA CRE ee Cag 19 
PLEIN ACOLALSAN ZCDIDETION ses Sere dc tite ure oe Bol vane 21 
SUAS ORR T STN RY ae ARG en ge RON Gir aa 22 
RUSE E ROA PIC ote Cl ts sy os Gee ek nels iene Se 24 
MIMI TAIVIGLTIPUICATION voi) .. des Fase a eh. eet 28 
wate 1S Me eee Ra ee 6, ee Ete a 29 
NP terg MSR ae Re in Fak pn ee ee 34 
AN TINGRLI UT NINES |.5 mk Oe ee eer as 42 
REIT ATES NON IWS 00) ee oss a ke Deeks os Bcd oe 45 
peepee MM ieee es ee Se Tn 46 
WUISRILMEIANL GON PY 3 Me te nt gilt. och ca ttc as oe 48 
mera es een ae sn ee ee ee SS 50 
| pe 1 ecg aE gees 7 ite all Sa lee A Ree a De 52 
ACID eA SUR Ey Bro ess cinta: Knot betes ioe ak 54 
APOC TW PEGE Ts se 6.25 oh os Kine ae we oe nahn aes 56 
PET TAT POPS EIGHT. bis br 28 muta oe vio te 58 
SPREE NITE TT Cee ote Sow tatn Pe deed 4 eget ie Swink Boaz 60 
MEERUT See kd OF oie Mics Sued he sv St Nall a ks Gah es Oe 62 
Sareea CN DA eer ee ee See. sho she oie gum ine 66 
errmm ee ter AND? IMISCOUNT). 2000s... Ge a5 «aires oreaie ss 70 
1 TW yn rete ote arp ae aa 72 
Leds og ES aoe 1s loo re SS icy: eM ea ee ee ces. 
PRISE PE Se A ATES crane eh ST eas aude ksi 4's eet 
SEY TIM eerie oe a pte Pe! BERS fakin, ei ws (Me 
et RAMEE eens, Sit Aree ee. hy Oy ae eee 80 
EUSIN ES. COMPUTATIONS. © baeevdios aie ene eel eeee 2 


MISCREITANEOUS, PROBLEMS ceo lccGis coca ea eees 82 


MANU-MENTAL COMPUTATION, 


NOTATION. 


1. Write numbers below one thousand in the order of 
“hundreds,” ‘‘tens,’’ “units.” 

Nore 1.—Students should not be allowed to enumerate 
by saying “units,” “tens,” “hundreds,” ‘thousands,” 
“tens of thousands,” ete. The fact that children begin 
with one and count to one thousand is no reason for 
learning the orders in the same direction, 

2. Give the fingers of the left hand the names of the 
periods; beginning with the thumb, call it trillions, the 
first finger billions, the second finger millions, the third 
finger thousands, and the fourth finger units—as per 
accompanying illustration (Fig. 2). 

Note 2.—To read and write numbers students always 
begin with the highest order and go toward units, so they 
should learn the periods in the order of trillions, billions, 
millions, thousands, units, and not as they usually do; 7. e., 
beginning with units. 

3. Each finger represents three figures, or the hundreds, 
tens and units of the period for which the finger is named: 
z.e., the first finger represents hundreds of billions, tens of 
billions and billions; the second finger represents hundreds 
of millions, tens of millions, millions; ete. ; 


4. Place the back of the left hand against the black- 
board. Spread the fingers far apart and keep them at 
even distances from each other. 

Note 3.—In this position a line drawn straight from the 
little finger should reach the decimal point; one from the 
third finger should come between thousands and hundreds; 
one from the second finger should come between millions 
and hundreds of thousands; ete, (Fig. 2.) 


12 MANU-MENTAL COMPUTATION. 


5. Write the one, two or three figures in each order 
before (to the left of) the finger which bears the name of 
that order; write the next lower order in the same 
manner, filling all vacant places with. ciphers. Con- 
tinue this process until the decimal point is reached. 

(N. B. Notation and Numeration of Decimals are ex- 
plained later.) 


6. Write: 
1. Six hundred forty-three. 
. Eight hundred seventy-eight. 
. Seven hundred eighty-five. 
. Eight hundred six. 
. Six hundred ninety-seven. 
. Sixty-two thousand four hundred eighty-three. 
. Seven thousand eight hundred ninety-two. 
8. Four hundred thirteen thousand two hundred - 
fifty-four. 
9. Seventy million forty thousand two hundred 
twenty-six. | 


10. Thirty-eight billion four hundred thirty-two mil- 
lion seventy-five thousand two hundred forty-seven. 


IQ oP Wb 


11. Two trillion seven hundred sixty-four billion three 
hundred eighty-two million seven hundred forty thousand 
six hundred eighty-three. 

12. Three hundred forty-two trillion seven hundred 
eight million sixty-five thousand seven hundred. 

13. Eleven trillion one hundred ten billion one million 
one hundred one. , 

14. Three hundred trillion six million twenty-two | 
thousand four hundred eighty. 


15. Nine hundred sixty-four trillion three hundred | 
five billion seventy-two million seven hundred thousand : 
four hundred sixty-two. gett 


42/. 


,2 45,209, 


627,486 


NOTATION 4no0 NUMERATION. 


14 MANU-MENTAL COMPUTATION, 
NUMERATION. 


7. To read a number, place the left hand with the back 
to the board or paper, in such a position that the fourth 
(little) finger points to the decimal point, the third finger, 
between the third and fourth figures (counting from the 
right end of the number), etc., so that three figures will be 
between each finger and the next higher used. 


Beginning with the left or highest number, read the 
figures to the left of each finger as a whole number, and 
add the name of the finger; continue this till the decimal 
point is reached, (Fig. 2.) 


Notre 4.—Have the student learn to write and read 
numbers on the blackboard, where the figures can be made 
large enough to place the fingers between the periods with- 
out crowding them, After this has been done thoroughly, 
the student can handle small figures on paper without 
difficulty. 


8. Read the following: 


1K 3-46. 16, 20342700321. 
2 478. Wy 146398462064. 
3 586, 18. 4482918000. 
4, PEO: 19; 1006070800989. 
5. 796. 20. 3860472834672. 
6 38462, 21. 789342876395825. 
7 29870. 22. 802341078639582. 
8 225316. 23. 304027894030. 
a 1420340. 24, 10001001101110. 
10. 7200408. 25. 27830472578334. 
il. 3046705. 26. 580008573095785. 
12, 12202007. 27. 90000087 366409. 
13. 28004962. 28. 4050307 65003. 
14. 665566488. 29. 57839567584. 


15, 704400096. 30. 7468000567008, 


MANU-MENTAL COMPUTATION. re 
CHECK READING. 


9. In many business houses where checking is done the 
numbers are read in the shortest manner possible. 

In reading numbers of three places, the word “hundred” 
is omitted and a short pause made between the hundreds 
and tens. 784 is read seven, eighty-four, instead of seven 
hundred eighty-four (7,84). 


In reading numbers of four places, the pause is made at 
the same place. 6791 is read sixty-seven, ninety-one, 
instead of six thousand seven hundred ninety-one (67,91). 


In reading numbers of five places, the pause is made 
between thousands and hundreds and between hundreds 
and tens. 36842 is read thirty-six, eight, forty-two, in- 
stead of thirty-six thousand eight hundred forty-two 
(36,8 ,42). 


In reading numbers of six places, the pauses are made 
between hundred thousands and ten thousands, thousands 
and hundreds, and hundreds and tens. 724936 is read 
seven, twenty-four, nine, thirty-six, instead of seven 
hundred twenty-four thousand nine hundred thirty-six 
(7 ,24,9,36). 

In checking, the check mark (7/),dash (—) and dot (.) 
are the marks usually used. Each should be used for a 
definite and distinct purpose. A good rule is to use the 
dot for numbers to be added (ledger, etc.) ; the dash where 
the numbers must be rechecked (recheck with a perpen- 
dicular line across the dash or a dot above it); and the 
check mark for final numbers and totals. 


Each good business house has a definite rule of its 
own, but there is no uniformity of usage. 


16 MANU-MENTAL COMPUTATION. 


RECORDING. 


10. In recording, the thumb represents tens of thou- 
sands; the first finger, thousands; the second finger, 
hundreds; the third finger, tens; and the fourth finger, 
units. (Fig. 3.) 

Note 5.—It will be noted that, in numeration and 
notation, the hand is spread and each finger represents 
three orders or a period, while in addition, subtraction, 
multiplication, and division, the fingers are not spread 
and each finger represents only one order. 


11. The end of the finger represents 1, 


the first ball 2s 
the first joint 3, 
the second ball 4, 
the second joint 5, 
the third ball 6, 
the third joint Fs 
the fourth ball 8, 
the crease in palm of hand 9, 


Nore 6.—It will be noticed that the ends and joints of 
the fingers represent odd numbers, and the balls represent’ 
even numbers. 

12. The thumb is numbered on front from 1 to 5, out- 
side 6 to 10, back 11 to 15, inside 16 to 20, and the base 
21 to 25, as per illustration (Fig. 3). In recording 16 to 
20 on the thumb, the thumb of the right hand comes 
between the thumb and the first finger of the left hand, 
with the right thumb nail against the left thumb. 


13. The thumb and fingers of the right hand are used 
as pointers to keep the record on the left hand. They 
are called ‘‘pointers.’”’ The record is made on the left 
hand, so it is called the “record hand,” and its fingers are 
called the “record fingers.” 


14. To record a number on the left hand, place the 
corresponding fingers of the right hand on the ioints or 


ADDITION. 
SUBTRACTION. 
/TULTIPLICATION. 
DIVISION. 


F/G d3 


18 MANU-MENTAL COMPUTATION. 


balls that represent the number of units in each order 
contained in the number. (Fig. 3.) (Position, Fig. 1). 
15. Illustration 1: 
Lecord 625. 

To record 625, place the 2nd finger of the right nand on 
the 3rd ball of the 2nd finger of the left hand (marked 
“6,” Fig. 3,), the third finger of the right hand on the 
first ball of the third finger of the left hand (marked 
2,” Fig. 3.), and the fourth finger of the right hand on the 
second joint of the fourth finger of the left hand (marked 
“5,” Fig. 3). Fig. 1 holds on record 58631. . 


Note 7.—Practice recording numbers by this method 
until it can be done quickly and accurately. It will be 
readily seen, by reference to Fig. 3, that 25 tens of 
thousands (250,000) can be recorded by this method. 
As comparatively few computations use such large 
numbers, the Manu-Mental method is sufficient for most — 
problems. (See note 12.) 


16. Record the following: 


1,, 42. Pie estilo: 21. 189645. 
2. 368. 12. 5682. 22. 212416. 
SS n2Uo, 15. (906; 23. 238044. 
4, 318. 14, 4045. 24, 256800. 
5. 406. 15. 4007; 25. 250400. 
6. 900. 16. 13640. 26, 178423. 
pero: 17. 25896. 27, 250074. 
8. 340. 18. 34276. 28. 116020. 
9. 676. 19. 80603. 29. 205790. 
10,192. 20, 20487. 30, 189500, 


MANU-MENTAL COMPUTATION. 19 


ADDITION. 


17. To add, record the first number on the left hand; 
add the next number by beginning with its highest order, 
moving the recording finger downward on the record 
finger (finger of the left hand) as many points as there are 
units in the digit in the order to be added: 


If this moves the recording finger past the crease in the 
palm of the hand, move the next higher finger one point— 
as in carrying 10—then return the recording finger to the 
end of the corresponding finger (marked ‘1’’) and con- 
tinue to move downward until all the units of that digit 
are used: 


Move thé next lower finger of the right hand downward 
on the left, in the same manner; continue this until all the 
orders have been added, at which time the sum will be 
recorded on the fingers. 


18. Illustration 2: 
Add 215 and 432. 


Record 215; move the second finger downward four 
points, to 6, the third finger downward three points, to 4, 
and the fourth finger downward two points, to 7, then 
the record is 6, 4, 7, or 647 for the sum, 


Note 8.—At first it is better to add numbers in which 
the sum of the digits in any column will not exceed nine: 
practice this until the result can be obtained quickly and 
the student has confidence in his results, 


19. Illustration 3: 
Add 321, 417, and 537. 


Record 321; move the second finger downward four 
points, to 7, the third finger downward one point, to 3, 
and the fourth finger downward seven points, to 8; then 
the record is 7, 3, 8, or 738. 


20 MANU-MENTAL COMPUTATION. 


To add 537 to 738, move the second finger down five 
points, to 12; (as twelve cannot be recorded on the second 
finger, record the one on the first finger, as in carrying 
ten, and the two on the second finger). Move the third 
finger down three points, to 6, and the fourth finger down 
seven points, to 15; (as 15 cannot be recorded on the 
fourth finger, record the five and move the third finger 
down one point, as in carrying 10, to 7), and the record 
stands 1, 2, 7, 5, or 1275. 

Note 9.—It is well to confine this operation to two or 
three numbers, of not more than three figures each, until 
the student has confidence in the method and is positive 
of the correctness of his results. After that the operation 


may be extended until the sum equals 250,000. The sum 
may be extended to 1,000,000, as explained in note 12. 


20. Add the following: 


A: 2. 3. 4, 
21 26 77 65 
33 42 20 21 
5. 6. ti 8. 
124 362 324 368 
232 324 572 684 
241 103 349 238 
2 10 Bl; 12 
26 897 685 768 
86 945 467 319 
92 530 268 864 
65 467 867 945 
78 970 905 833 
13 14 15. 16 
2634 3468 15421 78630 
3268 2567 18963 22340 
9473 4847 60325 90416 
2867 4862 31872 26590 


MANU-MENTAL COMPUTATION. 21 


MULTI-COLUMN ADDITION. 


21. Many bookkeepers add two, three or four columns 
at the same time. 

To add two columns, place the hand so one finger’ will 
come on each column just below the first number; draw 
the finger on tens column down below the next number; 
add the number thus exposed to the tens above; draw the 
finger on units column down the same distance; add the 
figure thus exposed to the units above, carrying the tens, 
if there are any, to the tens column; continue this until 
the fingers reach the bottom of both columns. 

To add more than two columns, place a finger on each 
column, draw the fingers down on ten-thousands, thou- 
sands, hundreds, tens and units columns, adding each 
figure exposed to the number already in its order. 


22. Add the following: 


i Ze 3. 

34 232 3487 
42 31] 2326 
33 416 3586 
27 534 9864 
96 896 7839 
74 347 5787 


Place the fingers and draw them down as directed 
above; after each move call the result as follows: 

23. Illustration 4: 34; 74, 76; 106, 109; 129, 136; 
226, 232; 302, 306. 

24. Illustration 5: 232; 532, 542, 543; 943, 953, 959; 
1459, 1489, 1498; 2293, 2383, 2389; 2689, 2729, 2736. 

25. Illustration 6: 3487; 5487, 5787, 5807, 5813; 8813, 
9313, 9393, 9399; 18399, 19199, 19259, 19263; 26263, 
27063, 27093, 27102; 32102, 32802, 32882, 32889. 


22 MANU-MENTAL COMPUTATION. 


SUBTRACTION. 


26. Record the minuend on the fingers of the left hand. 
Beginning with the highest order of the subtrahend, sub- 
tract it by moving the pointer (finger on the right hand) 
up toward the end of the finger as many points as there 
are units in the digit of that order. 


Subtract the next lower order in the same way, and 
continue this until “units’’? are subtracted, at which time 
the remainder is recorded. 


If moving any pointer upward the required number of 
points takes it off the end of the finger, move the pointer 
for the next higher order upward one point, counting this 
move as a point, then return to “nine” and continue to 
subtract as before. (This is equivalent to reducing one 
of the next higher order to ten of the lower order.) In 
many cases it is more easily performed by adding the 
complement of the figure to be subtracted and subtract- 
ing one from the next higher order. 


2/. Ilustration 7: 
Subtract 325 from 746. 


Record 746 on the left hand. Subtract 3 hundred 
from 7 hundred by moving the second finger of the right 
hand three points toward the end of the finger on the 
left hand; 7.e., to 4. Subtract 2 tens from 4 tens by 
moving the third finger of the right hand up two points; 
v.e., to 2. Subtract 5 units from 6 units by moving the 
fourth finger of the right hand up five points; 7.e., to 1. 
The record is 4, 2, 1, or 421 for the remainder, 


Nore 10,—A number of problems, where each figure in 
the minuend is larger than the figure in the corresponding 
order of the subtrahend, should be given for practice 
before attempting problems where it is necessary to 
reduce one from the next higher order. 


| 


MANU-MENTAL COMPUTATION. 23 
28. Illustration 8: 
Subtract 389 from 654. 


Record 654 on the left hand. Subtract 3 hundred from 
6 hundred by moving the second finger up three points; 
z.e.,to03. Subtract 8 tens from 5 tens by moving the third 
finger up five points (this takes it off the end of the 
record finger, so the second finger must be moved up one 
point, to 2), then take it back to 9 (which move counts 
one point, or six points moved) and move it up two more 
points, to 7. Subtract 9 units from 4 units by moving 
the fourth finger up four points (this takes it off the end 
of the record finger, so the third finger must be moved up 
one point, to 6), then return to 9 (which move counts one 
point, or five points moved) and move upward four more 
points, to 5. The record stands 2, 6, 5, or 265, 


29. Problems for Subtraction: 


1. 86—34, 16. 365 —72. 

2. 79 — 45. 17. 546 — 284. 

3. 59 —46. 18. 798 — 645. 

4. 88—57. 19. 916 —438. 

5. 96 —63. 20, 723 —536. 

6. 347 —316. 21, 1462—786. 

7. 794 —432. 22, 3794—1918. 

$2 927-715. 23. 6890 — 4972. 

9. 2468 — 1205. 24, 12612—7810. 
10. 7939 —4615, 25. 25604 — 6598. 
TL f2— 304, 26. 68509 — 18637. 
12. 61—46. 27. 90460 — 42300. 
13. 85—39. 28. 185640 — 38426. 
14. 78—59. 29. 210800 — 175050. 


15.5042 773 30. 238641 — 212968. 


24 MANU-MENTAL COMPUTATION. 


MULTIPLICATION. 


30. Multiply the figure in the highest order of the 
multiplicand by the figure in the highest order of the 
multiplier, and record the result on the fingers, leaving 
as many fingers to the right of the record as there are 
figures to the right of the two figures multiplied together. 
If the result exceeds nine, record the tens on the finger 
representing the next higher order. 

Multiply the next lower figure of the multiplicand by 
the same figure, recording the result and carrying the tens 
in the same manner as before. 

Continue until the entire multiplicand has been multi- 
plied by the figure in the highest order of the multiplier. 

Multiply the entire multiplicand, in the same manner as 
above, by the figure in the order next to the highest order 
in the multiplier, recording the results by adding them to 
that already recorded, 

Continue multiplying by the next lower order of the 
multiplier until the multiplicand has been multiplied hy 
the figure in units place in the multiplier, at which time 
the result is recorded on the hand. 


31. Illustration 9: 

Multiply 27 by 34. 

Three times 2=6. As there is one figure to the right of 
‘each of these figures, two fingers must be left to the right 
of the record. It must therefore be recorded on the 
second finger. (Record 600.) 

Three times 7 =21. As there is one figure to the right 
of 3 and none to the right of 7, there must be one finger 
to the right of the record. It must therefore be recorded 
on the third finger. As 21 cannot be recorded, the 2 
(which is really two tens of that order) is added to the 6 
already recorded on the second finger. (Record 810.) 

Four times 2=8. As there is one figure to the right of 


MANU-MENTAL COMPUTATION. 25 


the 2 and none to the right of the 4, there must be one 
finger to the right of the record. It must therefore be 
added to the record on the third finger. (Record 890.) 

Four times 7 =28. As there is no figure to the right of 
either the 4 or 7, the 8 must be recorded on the fourth 
finger and the two added to the third finger as above. 
This makes the record on the third finger more than nine, 
so the tens must be carried to the second finger and added 
there. (Record 918.) 


32. The above problem may be illustrated as follows: 
Multiply 27 by 34. 


Record 
3times 2= 6. Record onthe second finger, 600. 
“Se peed ay APA be ce Bucky a A sets bog 810. 
ee. O salar et CLOT s 890. 
Ae ee 28% _ rein LOUPEL TU oc 918. 


Note 11.—From the above it will be seen that this is 
ordinary multiplication, beginning with the higher orders 
instead of the lower, and recording on the fingers instead 
of paper or blackboard. 


33. Illustration 10: 
Multiply 346 by 67. 


Record 
6 times 3=18. Recordonthe first finger. 18000. 
GA a4 eben Se CONC 20400. 
6» -  6=—36. ot bb ec blvintl ¢ - Ga 20760. 
ane a eel, ¥ Tease OCU. 5 22860. 
7 “ 4=28. . fo retire: 23140. 
ji ie ae ee S ~~ & -1OUTt. - Past be wan 
34. Illustration 11: 
Multiply 486 by 397. 

Record 
3 times 4=12. Recordonthe thumb. 120000. 
Suet pee a 24 eek Ofiret, . finger, 144000. 


eee = Lo. oars, ssecond: . * 145800. 


26 MANU-MENTAL COMPUTATION, 


9 times4 =36. Recordonthe first finger 


Qe hs Bien FZ, # oe 2S Secon 
9 “ 6=54. - eatin - 
Y ti: ays. fy <  Ssecona= + 
Yt ~ ‘8'== 66. i =e Ftd = 
7 “ 6=42., . oe SST urtne oF 


35. Illustration 12: 
Multiply 3642 by 68. 


6 times 3=18. Record on the thumb. 


6... _6'=36. Pla sausllsteaciiiver 
647.9 424; 5 ee EC TROL tes 

Gus fae ele , iene EEL Tas 

Si aa eae Vio ee Fy) is 

8 “ 6=48. 2 . 2s SECON SS 

Se ee Fe. ‘ WGP L Dip ee naee 
87a a * ote SOUCGl! me 


36. Lllustration 13: 
Multiply 897 by 964. 


9 times 8 =72. Record 22 on thumb, 
(Drop 50. Remember 2. Note 12.) 
9 “ 9=81. Record on first finger. 
(Add 8 to 22 on thumb =30. 
Drop 25=5. Remember 3.) 


9 “ 7=63. Record on the second finger. 
6 “ 8=48. ee Se PATS - 
G7. Qa G4, Gee CO RECORU =: 
6 “ 7 =42 - “SS SEsird 7% 
4°“ §8=82. e Ee BECOUGs ae 
4 eee) =a Soe ee aie ae 
bee ts ee oy eed LOM te 


Record 

181800. 
189000. 
189540. 
192340. 
192900. 
192942. 


Record 

180000. 
216000. 
218400. 
218520. 
242520. 
247320. 
247640. 
247656. 


Record 


7220000. 


*51000. 

57300. 
105300. 
110700. 
111120. 
114320. 
114680. 
114708. 


MANU-MENTAL COMPUTATION. of. 


The record on the thumb is 11. To this must be added 
3 times 25, or 75. This makes the record 86 for the 
thumb. The result then is 864708. 


Note 12.—<As only twenty-five can-be recorded on the 
thumb, it will be necessary to drop the twenty-five when- 
ever the tens of thousands are greater than that. Call the 
twenty-five dropped one (which means once twenty-five) 
and remember it. Record the remainder on the thumb. 
By thus retaining 1, 2 or 3 in the memory, any numbers 
whose product does not exceed one million can be multi- 


plied. 


37. Problems for multiplication: 


ie ie V'4e es 15, ~ 90X98. 28. 236X362. 
2 24% 2. 16. 135 62. 29. ~ O18 K459: 
3. 12X4. 17. 386X399. 30. 865X675. 
4, 231 4. 18, 434X73. ol. 23467. 
5. 1214. 19. 862X 86. 32. 6057 X 4. 
6. 264. 20. 13846 X 26. 33. 7892X9. 
OS AT 21. 1347 X 34. 34. 4906X8. 
85> 89% 7. 22. 2130 34. 35. 34528 X 5. 
9. 136X8. 23. 5068 X 76. 36. 493739. 
10, 347X7. 24. 3900 49. Of, 1.1898 K 192. 


Litas 25.5 10; 25. 736X624. 38. 961X953. 
12, 47X82. 26. 438 623. 39. 928X904. 
13. 68X34. 27. 125% 234. 40. 997X988. 
14, 75X86. 


28 MANU-MENTAL COMPUTATION. 


MENTAL MULTIPLICATION. 


38. Students should be able to multiply all numbers 
below one hundred mentally. They can learn to do this 
and do it in much less time than they can learn the multi- 
plication table to twenty-five. 

To multiply two numbers of two figures each, multiply 
the tens together and call the result hundreds; multiply 
the tens of each number by the units of the other number, 
calling the results tens, and add them to the hundreds; 
multiply the units together, calling the result units, and 
add it to the result obtained above. This final sum is 
the product of the numbers multiplied. 


39. Illustration 14: 
Multiply 68 by 76. 
7<6=42. Call this hundreds, 
7X8 =56. Callthisteuns. 
Add and read by business method (p. 15) =47, 60. 
6X6 =36. Call this tens. 
Add and read by business method = 51, 20. 
8x6=48. Call this units. 
Add and read by business method =51. 68 or 5168. 


4. T]lustration 15: 
Multiply 97 by 82. 
89 =72 (hundreds). 
8X7 =56 (tens). Add =77,60. 
9x2 =18 (tens). Add =79, 40. 
7X2 =14 (units). Add =79, 54 or 7954. 


Notre 13.—This process is the same as multiplying on 
the fingers, but the partial results are retained in the 
memory instead of being recorded on the hand, thus 
making it mental instead of manu-mental. 


MANU-MENTAL COMPUTATION, 29 


DIVISION, 


1. Record the dividend on the left hand; remember 
the divisor. 

Find how many times the divisor is contained in the 
fewest left figures of the dividend. (This quotient figure 
may be recorded on the thumb or any finger, regardless of 
its order, not in use. When all are employed it must be 
retained by memory. Multiply each figure of the divisor 
by this quotient figure and subtract the results from the 
dividend figures used above. 

Using this remainder and the unused figures of the 
dividend as a new number, divide as before and record the 
result on the finger next to the one on which the previous 
result was recorded. 

Continue this division until the remaining part of the 
dividend is less than the divisor, at which time both 
quotient and remainder will be recorded on the fingers or 
retained in the memory. 


Norte 14.-—If in any division the quotient figure is made 
less than it should be, the remainder will be greater than 
the divisor; in which case it is not necessary to perform 
the operation again, as the error may be corrected by 
increasing the last quotient figure one, and subtracting the 
divisor from this last remainder. This may be done 
several times; or the quotient figure may be increased 2, 
3, or 4 and two, three or four times the divisor may be 
subtracted. 

Nort 15.—At first much memory will seem necessary, 
but practice will soon develop this so the average student 
can perform operations which will appear marvelous to 
the uninitiated. 


42. Illustration 16: 
Divide 423 by 8. 
Record 423 in left hand. 
42~+8=5. Record this on thumb. 
5X8 =40. 
40 from 42 =2 remainder. Recorded on third finger. 


30 MANU-MENTAL COMPUTATION. 


23+8=2. Record this on first finger. 

2X8 =16. 

16 from 23=7 remainder. Recorded on fourth finger. 

Then the record stands complete, with 52, recorded on 
the thumb and first finger (without regard to order) and 
7 recorded on the fourth finger. As 7 is the remainder 
and § is the divisor the result is 52%. 

43. Illustration 17: 

Divide 382 by 16. 

38-+16=2. Record on thumb. 

2X1=2. Subtract this from 3, which leaves 1. 

2 times 6 =12. . Subtract this from 1 and 8, or 18, which 
leaves 6. 

62+16 =3. Record on the first finger. 

3 times 1=8. Subtract from 6, which leaves 3. 

3 times 6=18. Subtract from 3 and 2 or 32, which 
leaves 1+. Then the record shows 2 and 3, or 23 with 14 
remainder = 2314 =237. 


Notre 16.—Where the divisor is 39, 57, 86, etc., use 
the next higher order of tens (or hundreds), 40, 60, 90, 
respectively, as the trial divisor for obtaining the quotient 
figure. If the quotient figure thus found is too small, it is 
easy to correct the error by the method shown in Note 14 
and Illustration 18, but if the quotient figure is too large 
the operation must be performed again. 


44. Illustration 18: 
Divide 4698 by 69. 

69 =nearly 70. (See note 16.) 

46+7=6. Record on thumb, 

Multiply quotient figure by divisor. 

6 times 6=36. Subtract from 46 (recorded on Ist 
and 2nd fingers) and 10 remains (1st and 2nd fingers). 

6 times 9 =54. Subtract from 109 (1st, 2nd and 3rd 
fingers) and 55 remains (2nd and 3rd fingers). 


MANU-MENTAL COMPUTATION, 21 


As the remainder, 55, is not equal to the divisor, 69, the 
quotient figure, 6, must be correct. 


55+7=7. Record on Ist finger. 


7 times 6 =42. Subtract from 55 (2nd and 38rd finger) 
and 13 remains (2nd and 3rd finger). 


7 times 9 =63. Subtract: from 138 (2nd, 3rd, and 4th 
fingers) and 75 remains. (3rd and 4th fingers.) 
Result, 67 with a remainder of 75. 


As the remainder (75) is greater than the divisor (69), it 
is evident that the last quotient figure is not large enough. 
Correct the error by adding 1 to the last figure of the 
quotient (7+1=8) and subtracting once the divisor (69) 
form the remainder (75), leaving 6 for the remainder. 


The final result is therefore 68 with 6 remainder cr 
68.85 = 6875: 


45. Illustration 19: 

Divide 7642 by 37. 
76+37 =2. Retain in memory. 
2X37 74; Subtract from 76 =2 remainder. 


As 24 carinot be divided by 37, it is necessary to include 
the two additional orders before dividing again and the 
quotient obtained from this division must occupy two 
orders. As no quotient figure is more than 9, the second 
order must be filled with a naught. 


242 + 37 =6. (Put a cipher before it and retain in 
memory.) 


6 times 3=18. Subtract from 24 (2nd and 3rd fingers) 
=6. : 


6 times 7 =42: Subtract from 62 =20. 


Result, 106 with-20 remainder or 106328. 


32 


46. Illustration 20: 
Divide 168482 by 338. 


1684 +338 =4. Remember. 
4X 338 subtract from 1684 =332. 


MANU-MENTAL COMPUTATION. 


3328 +338 =9. 


9X 338 subtracted from 3328 =286. 


2862 + 338 =8. 


8X 338 subtracted from 2863 =158. 


Result 4, 9, 8; with 158 remainder, or 498438, -o: 


47. Problems for division: 


. 83+6. 


2. 93+7. 


49 +3, 


. 168+9. 


725-8. 
871+1. 
687 +8. 


. 482+7. 

. 3845178. 
. 2789 +8. 
. 7003 + 2. 
, OOLoarD: 
. 8427 +9. 
. 9304 +6. 
Pf o5 Fle 
. 865+ 24, 
. 658 + 27, 
. 469 +65. 
. 802+ 72. 
. 105+19, 


TAN 


22 


am, 


23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
3l. 
32. 
33. 
34, 
35. 
36. 
37. 
38. 
39. 
40. 


4678 + 26. 
2576 + 48. 
3400 + 93. 
6075 +77. 
3864 +61. 
9007 + 132. 
6803 + 365. 
2345 + 538. 
5097 + 960. 
2045 + 654. 
23021 +6. 
35680+9. 
97342 + 34. 
78623 + 87. 
60035 + 136. 
72038 + 309. 
30856 + 568. 
79832 + 974, 
58967 + 2130. 
98893 +8725, 


MANU-MENTAL COMPUTATION, 33 
AS. Review Problems: 


1. I traveled in the train six days; the first day the 
train traveled 345 miles; the second, 294; the third, 332; 
the fourth, 367: the fifth, 392; and the sixth, 416. How 
far did I travel? 


2. I bought three pieces of property: the first cost 
$3467; the second, $9688; the third, $325 more than both 
the first and second. How much did all cost? 


3. On balancing my books at the end of the year,-I 
find our firm has paid the following expenses: rent, $5786; 
light, $470; drayage, $1689; insurance, $950; fuel, $2306; 
clerk hire, $3875; sundries, $1962. The goods cost $32578 
and sold for $64855. What is the firm’s net gain or loss? 


4. I bought 12 cows at $22 each, 4 horses at $65 each, 
124 hogs at $7 each, and 388 sheep at $3 each. How 
much did I pay? 


5. I sold 472 cattle at $46 a head. After depositing 
$8685 in the bank, I purchased a lot for $650, built a house 
on it for $9827 and bought 34 acres of land with the re- 
mainder. Find cost of one acre. 


6. Sold four pieces of property for $3975, $6240, 
$12600 and $14485. I deposited one third of it in the 
bank and divided the remainder among my 5 children. 


How much did each receive? 
es 


7. ILexchanged 13 barrels of molasses, each containing 
32 gallons, at 50 cents per gallon, for 25 bolts of cotton 
worth 8 cents a yard. Find how many yards in a bolt. 


34 MANU-MENTAL COMPUTATION. 


DECIMALS. 

49. In notation and numeration of decimals, the hand is - 
used to keep the periods named and separated just as in 
notation and numeration of whole numbers. (Figs. 4 and 5.) 

50. Notation of Decimals. 

Write the decimal point and call it by the name of the 
denominator of the decimal fraction. 

Place the left hand so the finger, which bears the name 
of the period in which the denominator is found, will be 
to, the right of the decimal point, so the remaining orders 
of that period, if there are any, may be written between. 

If the next. lower order contains figure, other than zero, 
write the numerator of the fraction as a whole number. 

If the next lower order contains no figure, other than 
zero, fill each place with a cipher until the highest order of 
the numerator is reached, then write the numerator as a 
whole number. 

Note 17.—By this method of writing decimal fractions 
the decimal points, being written first, can be placed under 
each other and the decimal fractions written in position 
for adding or subtracting, and the student will grasp the 
idea of the decimal fraction much easier and be able to 
handle it much more readily than he will when compelled 
to write the numerator as a whole number, then begin 
with “tenths” (at “units’”) and enumerate to the left, 
filling vacant places with ciphers, to place the decimal 
point. 

51. Write: 

Seven hundredths. 

. Twenty-eight hundredths. 

Thirty-five thousandths. 

. Four hundred five thousandths. 

. Nine thousandths. 

Seven hundred two ten-thousandths. 

. Ninety-five millionths. 

. Seven thousand four hundred sixty-eight tril- 


lionths, 


ME OT BR OO ND 


CO 


FIG. 
36,453,247,04,628, 


/ ‘ 


7 
ee 


FIGURE 5. 
DECIMALS. 


36 MANU-MENTAL COMPUTATION. 


9. Eighty-seven ten-millionths. 

10. Three hundred fifty seven hundred-millionths. 

11. One million seven hundred thousand sixty-nine 
ten-billionths. 

12. Three hundred sixty thousand two hundred three 
millionths. 

13. Eighty-eight and five thousand five hundred- 
thousandths. 

14. Seven, and seventy-five thousand five hundred- 
thousandths. 


15. Three hundred six, and thirty-five ten-thou- 
sandths. 

16. Forty thousand seventy-six, and twelve hundred- 
thousandths. 

17. Six, and one thousand six hundred forty-two 
ten-millionths. 

18. Thirty-seven, and thirty-seven thousandths. 

19. Twenty-five million, and twenty-five millionths. 

20. Six hundred thirty thousand, and sixty-three 
thousandths. 


52. Numeration of Decimals: 


To read a decimal fraction place the hand beneath the 
fraction, so the fourth finger will come to the right of the 
‘units’? and the third finger so three orders will come 
between it and the fourth finger, etc. (Fig. 4.) Read 
the figures of the fraction as if they were a whole number, 
then call the denomination by the order which the decimal 
occupies, adding “‘ths’” (hundred, hundredths, ete.). To 
read a whole number and decimal: Read the whole number, 
then move the hand as in Fig. 5 and read the decimal. 

Norte 18.—In this method the decimal point is consid- 
ered as occupying an order just as a figure does. It is not 
necessary to ‘“‘numerate from the decimal point” nor to 


“begin with the right hand figure and call it tenths” to 
numerate s read the decimal fraction. 


MANU-MENTAL COMEUTATION, oe 


53. Read: 


1. .0047 
2. .3406 

3. 9.326 

4. 25.0003407 

5. 9230010. 04 

6. .00000300007 

7. 3708 .9200700006 

8. 200000. 000002 

9. 700800.800007 

10. .003% 

11. 400.733 

12. 6.0035782 

13. .0008 

14. .60072 

15. 58774862.57915060 

16. 8005076000. 005000347705 

17. 3794.0000005789456 

18. .000000000475603 

19. 2876435 0042001056783 

20. 479346783257862 .56791132782462 


54. Addition of Decimals: 
Decimals may be added as whole numbers are added. 


In adding numbers that contain both whole numbers 
and decimals, it is well to spread the fingers between 
which the decimal point should come and use each finger 
to represent an order (Figs. 6, 7, 8, 9.) 


Nore 19.—In reading and writing numbers, the fingers 
are spread and may cover five periods or fifteen figures; 
but in addition, subtraction, multiplication and division, 
the fingers are not spread and cover but five orders (five 
figures), 


38 MANU-MENTAL COMPUTATION: 


55. Subtraction of Decimals: 


Remember the instruction for subtraction of whole 
numbers and apply the suggestion given under addition of 
decimals. 


56. Multiplication of Decimals: 


Multiply, decimals just as whole numbers. 

When two figures were multiplied together they were 
recorded on the fingers, leaving as many fingers on the 
right of the record as there were figures on the right of the 
figures multiplied together; so in multiplying decimals 
there will be as many figures to the right of the decimal 
point as there were to the right of both decimal points. 


57. Division of Decimals: 


Divide decimals just as whole numbers are divided. 

Find the difference in the number of decimal places in 
the divisor and dividend. If those in the dividend exceed, 
point off that many decimals in the quotient; if those in 
the divisor exceed, add that many ciphers to the result. 


Note 20.—In division of decimals by the ordinary 
method (with paper and pencil), when the divisor is mul- 
tiplied by a figure of the quotient the decimal point should 
be placed i in each result until it comes beneath the decimal 
point in the dividend, at which time the decimal point 
Should be placed in the quotient. 

Tf the divisor is larger than the dividend, divide the 
fingers as in Figs. 6 to 9, to show the number of places in 
the divisor; place the hand beneath the dividend, with the 
opening between the fingers beneath the decimal point, at 
the same time placing the decimal point in the quotient; 
move the hand to the right, placing a cipher in the quo- 
tient each time the hand is moved one order, until the 
divisor may be subtracted from the figures in the dividend 
directly above it, and then divide as in whole numbers. 
Many students who seem unable to understand division 
of decimals will become prompt and accurate by this 
method. 


DECIMALS. 


40 MANU-MENTAL COMPUTATION, 


58. Problems for Addition of Decimals: 


ib 6. 
~Q75 3.25 
- 308 20.3 
-o0 35.68 
ces 78 .05 
1257 
2 7. 
.037 2.205 
.0469 3.065 
.3278 9.12 
.0598 4. 
eal, 7.653 
6.003 
3.07 
3. 8: 
.00653 1.0003 
.0398 4.7006 
.6372 5.04 
.97503 2.057 
(8932 3.1028 
4. a: 
.457 128.79 
. 2587 346.3 
. 3698 (2La 
.0598 35.25 
60.32 
9.4 
5 10. 
00005 10. 
03 13.1 
6007 7.4 
4603 136.9 
90876 230 
0057 416.3 
0003 350.4 


MANU-MENTAL COMPUTATION. 41 


59. Problems for Subtraction of Decimals: 


PAP sear h a o 


. 784 — .036 
.58 — .026 

. 7432 — .065 
. 2879 — .0934 
.4— .0349 


6. 


(P 
8. 


9. 
10. 


4.25—2.06 
12 UL oO 
120 — .012 
1.005 — .697 
3.016 — 2.697 


60. Problems for Multiplication of Decimals: 


2.6X7 

.39X1.5 
.07 X .34 
7.8X34 
18 X .34 


db kd 
. 30.4X .26 

. 13.4 26 

. 6.238 X .4 
. 12.5 .006 
. 130X .072 
. 325 X 680 
. 75.8 X12 

. 3.47 X 203 
. 20.5 .036 


16. 
17. 
18. 
19: 
20. 
21. 
22. 
23. 
24, 
25. 
26. 
27. 
28. 
29. 
30. 


378 X1.034 
622 x .006 
70.3 X .00009 
90000  .0065 
.0004 X . 2648 
5.065 X68 
BOT O55) 

.054 X43 
.00075 X7.38 
.0412X .00027 
eh Sa 
tS x IS x OLS 
7X 76X .706 
.018 X .034X .047 
46X4.06*4.6 


61. Problems for Division of Decimals: 


L 


4 


me 


3. 
4. 
5. 


fon) 


43.60 + 52 
9.68 +6.4 
.065 + 78 
.05475 + 15 

. 11928 = .056 
.04905 + 237 
.00594 + .039 


. 126.54+7.03 
. 1.2288 +51.03 
. 190+ .038 


LF: 
12. 
133 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


46 + .0004 
.46+ 40 
4.6+4 
164+2.05 
.0027 + 67.5 
.00065 + .8125 
. 0683 = . 009 
81+4.05 

.016 +160 

. 20002 + 400. 04 


42 MANU-MENTAL COMPUTATION. 


CASTING OUT THE NINES. 


62. The’student should test results by casting out the 
9s. This is not a positive test. It will not indicate an 
error of nine or any multiple of nine, but it will indicate 
any other error, and is therefore valuable. 

Every number has a “test figure.” It is obtained by 
casting out the 9s. This is done in two ways: Ist, by 
adding the digits and dividing by 9; the remainder is the 
test figure: 2d, by adding the digits together continu- 
ously until the result has but one figure in it. 

63. Illustration 21: 

Find the test figure. for 346. Add 3+4+6=13. 
Divide by 9=1 with 4 remainder. The remainder 4 is 
the test figure. 

64. Illustration 22: 

Find the test figure of 4867. Add 4+8+6+7 =25. 
Divide by 9 =2, with 7 remainder. 7 is the test figure. 

65. Illustration 23: 

In the above illustration, the 9s may’be dropped at 
any time as follows: 

Find the test figure for 4867. Add-4+8=12. (As 
this is more than 9, subtract 9 from it). 12—9=3. 
3+6=9. 9-—9=0. 0+7=7. 7 is the test figure. 

66. Illustration 24: (Second method.) 

Find test figure for 176. Add 1+7+6=14. 

Add these digits together. 1+4=5. 5is the test figure. 

67. Illustration 25: ‘ 

Find the test figure for 6879467. Add6+8+7+9+4+ 
64+7=47. 447=11. 14+1=2. 2 is the test figure. 

68. To test addition by casting out the 9s, find the test 
figure for each number, then find the test figure for these 
figures. If the test figure for the sum is the same, the 
addition is correct unless a mistake of 9 or some multiple of 
9 has been made. 


MANU-MENTAL COMPUTATION. 43 


69. Illustration 26: 
Add. Test figure: 


426 3 
347 5) 
819 0 
240 6 
674 8 
2506 22 


2+5+0+6=13. 1+3=4. Test figure. 
2+2=4. Test figure. 


70. To test subtraction by casting out the 9s, subtract 
the test figure of the subtrahend from the test figure of the 
minuend. The result will be the test figure of. the re- 
mainder. 

If the test figure of the minuend is smaller than the test 
figure of the subtrahend, add 9 to it and subtract. 


71. Llustration 27: 


Subtract. Test figures. 


4732 tt=7) 
3684 21 = 3, 
1048 | 4. Test figure 


1+0+4+4+8=13=4.. Test figure. 


72. Illustration 28: 

Subtract. Test figures. 
78409 28 =10=1. As 2 cannot be subtracted 
47693 29=11=2. from 1, add 9 to the 1, 
——— ————— whichmakes10. Subtract 
30716. 2 from 10, which equals 8. 

Test figure is 8. 
38+0+7+1+6=17=8. Test figure, 


44 MANU-MENTAL COMPUTATION. 


73. To test multiplication by casting out the 9s, find 
the test figure for the multiplier, and for the multiplicand. 
Multiply these together. The test figure of this result 
will be the same as the test figure of the product of the 
multiplier and multiplicand. 


74. Illustration 29: 
Multiply. Test figures. 


137 11 =2. 
42 =6. 

5754 12. 

5754 =21 =3. 


12=3. Then the product is probably correct. 


75. To test division by casting out the 9s, multiply the 
test figure of the quotient by the test figure of the divisor 
and add the test figure of the remainder. The result will 
be the test figure of the dividend. 


76. Illustration 30: 
Divide 4623 by 5. 
5)4623—15=16. Test figure. 


924—3 remainder. 
5X6 =30. 30+3=33=6. Test figure. 


77. Illustration 31: 
Divide 3416 by 23. 
23)3416(148. Test figure 4. 


23 23. Test figure is 5. 

—- 3416. Test figure is 5. 

111 4X5=20. 20+3=23<65, 
92 Test figure. 
196 
184 


12. Test figure is 3. 


MANU-MENTAL COMPUTATION. 45 


UNITED STATES MONEY. 


78. In problems containing dollars and cents, spread the 
second and third fingers (Fig. 7). The opening between 
them will represent the decimal point. Record the dollars 
on the thumb, first and second fingers. If problems con- 
tain dollars only, use the fingers as in whole numbers. 

To reduce dollars and cents to cents, close the opening 
between the second and third fingers and the record is cents. 

To reduce cents to dollars, make an opening between 
the second and third fingers and the record is dollars 
and cents. 


79. Problems in United States Money. 
1. Add: $134.20, $75, $68.30. 


uly « $2460, $4725, $396. 
+ es $46.35, $47.80, $265.60. 
7 og al $125.35 $78.93, $734.60. 
i> = $34.95, $47.86, $79.65. 
6. Subtract: $76.85 from $285.20. 
Gs : $92.47 “ $107. 
8. G $17.87 “ $32.60. 
9. = $34.40 “ $67.95. 

10. : $76 “$134.20. 

11. Multiply: $32.26 by 6. 

ee te $9.47 by 15. 

js $18.65 by 46. 

14, : $.75 by 132 

eS $3238.60 by 7. 

16. Divide: $8.37 by 16. 

ibs ‘ $78.60 by 30. 

tthe $2647 by 36. 

19. “  - $147.80 by 127. 

20. $9 by 27. 


Notre 21.—In problems like the 15th, multiply the 
cents first, remembering the result, then use all the fingers 
in multiplying the dollars. 


Notr 22.—Canadian money is treated just the same as 
United States money is treated. 


46 MANU-MENTAL COMPUTATION, 
FRENCH. MONEY. 


80. French money is used in Switzerland and Belgium. 

The lira of Italy, peseta of Spain, drachma of Greece, 
and the bolivar,of Venezuela are the same as the frane 
(19.38¢c). 

To reduce francs to dollars of United States money 
multiply by .193; or for approximate value divide by 5. 

To reduce dollars to franes divide by .193; or for approx- 
imate value multiply by 5. 


81. Problems in French Money: 


1. Add: fr. d. ¢c m. 
30 es 4 0 
13 0) 6 5 
so pe 3 AG 8 7 8 
25 it 0 Gis 
oD od ake fr. d. Cc. m. 
10 3 4 0 
8 7 9 6 
15 2 0 7 
20 7 6 3 
3. Subtract: fr. d. C m. 
eye: 6 3 4 
6 8 2 9 
4, a ii d. C; m. 
38 0 0 6 
16 a7 9 8 
5. 2 iT: d. Gs m, 
76 0 0 0 
38 vi 3 5 
62 Multiply =. 7ir, 33 Sd.eUG, 7m. by 9 
fe e 2h Odes osc, (AG 
8. Z 135ir £44; 4 2g A 
9. Divide: l7ir,) 24d, See ber. 
aA WE? Pag 135in, 4920. = 28: 
11. Reduce: $25.00 to franes (exact). 
Le . $30.40“ - “ cee 
13. ‘. $120.00“ “ (approximate). 
14. 2 $78:35 “> =“ - 


aL. “ $9.65 “ “ “ 


. 


2 / 
F/G. 10. 
FRENCH MONEX. 


48 


MANU-MENTAL COMPUTATION. 


ENGLISH MONEY. 


82. To reduce £ to $, multiply by 4.8665, or for approxi- 
mate value multiply by 5. 

To reduce $ to £, divide by 4.8665, or for approximate 
value divide by 5. 


83. Problems in English Money. 
1. Add: 


. Subtract: 


“ 


. Divide: 


“cb 


. Reduce: 


“ 


. Multiply: 


£ S. d. far. 
12 14 7 
20 10 3 1 
16 18 4 a 
8 0 9 0 
£ s. d. far. 
35 16 11 Ms 
20 10 5 1 
8 15 f; - 
4 0 3 
“5, S d. far 
25 9 10 2 
12 19 4 Je 
oe s d. far. 
146 33 6 2 
68 9 2 3 
£ s d. far, 
216 0 0 
187 13 7 2 


£6 4s. 6d. 3far. by 7. 
£4 17s. 4d. 2far. by 24. 
9s. 6d. 3far. by 137. 
£26 3s. 10d. 2far. by 7. 
£135 7d. by 16. 

£12 J4s. 2far. to far. 
£120 10s. tod. 


ENGLISH [TONEY 


MANU-MENTAL COMPUTATION. 


13. Reduce: 


14. 
15. 
16. 
47. 
18. 
19. 
20, 
21. 
22. 
23. 
2A, 
25. 


240d. to £ 
646d. to £ 
785far. to 
3808s. to £. 


s. d, 
s. d. 


integers. 


1580d. to integers. 
6430far. to integers. 
620far. tos. 


13532 to £ 
£8 to 
AGS 
cody 1) Boe 
£34 * 


. 


$ (exact). 

$ “ 

$ “ 

$ (approximate), 
$ ““ 


GERMAN MONEY. 


84. 100 pfennigs make one mark (Reichsmark), 

For German money use the hand the same as in U.S. 
money. The thumb, Ist and 2d fingers represent 
marks, while the 3d and 4th fingers represent pfennigs. 

The mark is equal to 23.8 c. in U. 8S. money. 

To reduce $ to M. divide by 23.8, or for approximate 
value divide by 5 and add ¢ of the result to it. 

To reduce M. to $, multiply by 23.8, or for approximate 
value divide by 5 and subtract $. 

In rough estimates the Mark is considered 25c, or 48. 


85. Problems in German Money: 
8.24 M. to $ (approximate). 


> 
2. 
3. 
4 
5) 


e 
. 


6. 
fi 
8 


Reduce: 


15 M. to$ < 
$9.20 to M. ¥ 
$35 to M. * 


7.34 M. to $ (exact). 
30 M. to$ ¥ 


$10.35 to 
$28 to M. 


M. “ 


“ 


MANU-MENTAL COMPUTATION. 51 
86. Review Problems: 
1. What is the cost of a bill on Paris for 1346 francs? 


2. What is the cost of a bill on Amsterdam for 3426 
marks? 


3. What is the cost of a bill on London for £12 10s. ? 


4. Bought 10,000 pounds of dressed beef at 8 cents 
per pound, shipped it to France at a cost of $250 and sold 
it at 8 decimes per pound. How many dollars did I gain? 


5. Bought a bicycle in London for £5 and sold it in 
Madrid for 150 peseta. Did I gain or loose and how much, 
in United States money? 


6. I can buy a suit of clothes for 45 francs in Paris, 
£2 in London, 33 marks in Berlin or $28 in New York. 
Which is the cheapest place to buy it and how much 
cheaper than each of the other places is it? 


7. I traveled in Europe last summer, spending $75 
for steamer, etc., £40 5s. in England, 325 francs in France 
and Switzerland, and 116 marks 50 pfennigs in Germany. 
How much did my trip cost me? 


8. I had $1200 when I left New York. I spent 7 
of it in London, + in Germany, + in Belgium, 34 in 
Italy and 1 in Switzerland. How much, in native coin, 
did each country receive? 


9. A German owes an Englishman 20 marks, he gives 
an order on a Frenchman for 15 frances and pays the rest 
of the debt in English money. How much English 
money does he pay? 


Sr 


4 MANU-MENTAL COMPUTATION, 


DRY MEASURE. 
87. Problems: 


1. Add: Due nice are ts pt. 
12 3 4 1 
13 2 3 1 
68 2 0 1 
37 0 6 0 
2 + bubte pkee=s0t pt. 
10 y 5 0 
6 3 + 1 
5 2 0 0 
3 0 0 1 


3. Subtract: bu. pk. qt. pt. 


8 1 5 0 

4 . bu pk. qt pt 
120 2 3 0 

34 3 7 1 

5 : bu. pk qt pt 


6. Multiply: 35 bu. 2 pk.4 qt. 1 pt. by 7. 


8 : 16 bu. 3 pk. 6 qt. by 18. 
8. Divide: 28 bu. 2 pk. 7 qt. 1 pt. by 4. 
Igr 125 bu. 3 pk. 2 qt.1 pt. by 235. 


10. me 76 bu. 2 pk.5 qt. by 235. 


4 

SS 
6 
Pte 


Dk. 
1 
2 
3° 
1 
2] 
a) 
FIG. 12 
DRY MEASURE. 


54 MANU-MENTAL COMPUTATION. 


LIQUID MEASURE. 
88. Problems: 


1. Add: brl. gal. qt pt. ol, 
P rH 2 1 2 
3 18 0 1 3 
i 28 3 0 3 
2 5 : 1 0 
Da ie bri, + gal; qt. pt.. gt. 
4 7 2 1 1 
3 30 0 1 0 
1 31 2 it 3 
3) 2 5) 1 2 
a 0 3 0 1 
3. Subtract: bri. gal. qt. pt ei. 
5 18 3 1 3 
1 7 2 1 Z 
y nie bri. | tal Gis =) pee ae 
16 v 2 0 1 
4 8 0) 1 3 
bios bri esl Se ob pt. i 
9 0 0 0 
2 26 2 1 2 


6. Multiply; 2 brl. 7 gal. 2 qt. 1 pt.3 gi. by 4. 


ri 7% 5 bri. 12 gal. 3 qt. 2 gi by 35. 
&. Divide: 4 br]. 3 gal..3 qt 1 pt. 2 gi by 7. 
9. > 15 brl. 26 gal. 3 qt. 1 gi. by 18. 


10. « 99 brl. 24 gal. 2 qt, 1 pt. 3gi. by 46. 


F/G. J3A. 


56 


MANU-MENTAL COMPUTATION. 


APOTHECARY WEIGHT. 
89. Problems: 


1. Add: tb ¥ 3 se) gr 

3 5 f 2 ¢ 

6 a 2 1 12 

age | 6 1 Aid 

Ps ri 3 2 5 

Ome ib 5 5 3 er. 

4 0 3 1 6 

5 fe 2 0 8 

3 9) 0 2 14 

3 6 0 2 0 

Oe 3 1 16 

3. Subtract: Ib 3 3 KS) er. 

12 6 4 2 ‘4 

8 3 2 1 4 

4. “ tb 35 5 oe) er, 

10 9 2 0 6 

+ 3 5) 2 3 

Gr tb 3 3 Oe at: 

22 0 6 3 + 

7 3 7 0 16 

6. iS tb 3 3 pe or 

13 0 0 0 0 

2 6 ‘i 1 18 

7, Multiply: 33 1b.6 3 43 195 2ear. by 6. 

8. 4 21b.03 73 2B 15gr. by 26. 
9. Divide: 4ib.73 33 19 10 ar. by 26. 
10 8ib.75 63 2D 14 er. by 25. 


FIG. 14. 
APOTHECARY WT. 


58 MANU-MENTAL COMPUTATION. 


AVOIRDUPOIS WEIGHT. 


90. Reduce pounds to T., ewt, Ib. Record the 
pounds on the fingers; spread the first and second, and the 
second and third fingers (Fig. 16). Divide the number 
recorded on the thumb and first finger by two. If there 
is a remainder carry it to the next finger, where it will add 
10 to the record: 


91. Problems. 


1. Reduce: 15478 lb. to T., ewt., Ib. 
ore ‘i 26475 lb. to T., ewt., Ib. 
Oe ‘ 5 TL, 16 cwt., 75bato ib, 
4. 5 65 T., 3 cwt. to lb. 
Be * 6543 Ib. to ounces. 
5 ot. Le 12 ewt.23 or. 6 1b) 10 eto oe 
7 28930 oz. to integers, 


_~ 


8. Add Tee CW i Ul eee 
3 5 0 20 12 

ee 18 2 23 8 

3 5) 2 20 fi 

5 2 1 16 14 

9, Subtracte-l, > ewi./ qr. 1b: OZ. 
8 6 22 4 

> 3 9 5 

10. “* DE cutee saree ia aeons 
12 2 1 5 9 

4° te Dan. Sire wet 


11. Multiply: 2 T..7 ewt.-4*qr.-16 tb. 5 ez. -by 5: 

12. ‘ 1 T. 716 cwt.. 3 gr. 10%b, 3 oz. By AG. 
13.. Divide: 2 T. 7 ewt. 1 qr.-20)1b, 1062; by 6, 
14. . 5 T. 16 ewt. 3qr. 7 lb. 10 oz by 34. 


cut. 4b, 
4 2 24 i 
2 \2 22/2 
13 Se 23/-9. a 
aes pee “44 2/7 
*\,. Set LS as 
ce Ot oe ar ‘Sv 
Lv 8 6 ‘Ay 
* SOs 3a 
y %) Soe, go *% 
DB - 19> 8 | 
0 Ae 
nt ° 10 0 
21 10 
Ns) 
22 
Oe) 
3 = ® 
FS Glo: 


AVO/RDUPO/S WT. 


FIG. [0.. 
AVOIRDUPOIS WT. 


60 


MANU-MENTAL 


TROY =W HIGHS 


COMPUTATION, 


92. Problems: 


1. Add: 


“ie DWis 2 ot 
17 ie 16 22 
4 8 6 10 
3 5 7 12 
ff 10 19 6 
PRES lb. oz. pwt. gr. 
7 0 4 17 
10 6 + 12 
3 0 16 4 
4 9 14 3l 
3. Subtract: Ib. OZ —SDWte oy. 
135 10 18 val 
67 8 Queers 
4, lb. Gz.. (i pwiewer: 
86 2" + 12 
62 Sara kh yee 
5. : ib.2""02" 5 spwite aver. 
76 0 0 0 
25 10 4 14 


6. Multiply: 8 lb. 4 oz 
ie « 16 lb. 9 oz 
. Divide: 


.7 pwt. 12 gr. by 8. 
.14 pwt. 17 er. by 28. 
. 16 pwt. 4 gr. by 7. 
.10 gr. by 35. 


8 7 1b.4 07 
9. : 18 lb. 8 oz 
10. i‘ 135 lb. by 246. 


62 


MANU-MENTAL 


LONG MEASURE. 


COMPUTATION, 


93. In most practical problems in Long Measure, the 
higher numbers, miles and rods, or the lower numbers, 
rods, yards, feet and inches, are used. 


one or the other hand will answer. 


In either case 


(Figs. 18 and 19.) 


In yards record the half on side of finger. 
94. Problems in Long Measure: 


1; Ada: 
2. " 
3. : “ 


4, Subtract: 


rd. $= vide ft. in. 
26 2 1 7 
6 4 2 9 
3 4 0 3 
12 5 2 11 
mi, rd. 
12 120 
24 72 
35 . 307 
11 135 
29) I a yd. ft. in. 
12 160 2 1 7 
2S pilose + 2 8 
12 | 247 0 1 10 
3 78 5 2 6 
mi rd. 
| 
8 130 
rd. yd. ft. in. 
26 2 2 8 
16 4 2 10 
mi. 8 Rip tee aa ft. in. 
12 18 2 1 6 
4 35 5 2 9 


| F/G. 18. 
LONG MEASURE. 


FIG. /9 
LONG NEASURE. 


64 MANU-MENTAL COMPUTATION. 


Fb Multiply; 7rd.4 yd. 2 ft.8 in. by 7. 


8. 3 rd. 2 yd. 1 ft. 4 in. by 36. 
9, f 6 mi. 147 rd. by 28. 
10. "i 4 mi. 65rd. 4 yd. 2 ft. 7 in. by 9. 


11. Divide: 7rd.4 yd. 1 ft. 6 in. by 7. 

12. e 15rd. 3 yd. 2 ft. 8 in. by 46. 

13. ‘. 4 mi. 235 rd. by 26. 

14. 23 2mi.116rd. 2 yd.t-ft. Sm. by 9._ 


Nots 23.—In problems like the third, sixth, tenth and 
fourteenth, either the higher or lower denomination must 
be remembered, as five fingers are not enough to record 
them on. If the student has been thoroughly drilled in 
the previous work, he can work these and all similar 
problems by the Manu-Mental process. 


95. Review Problems: 


1. I bought four pieces of cloth, 5 yd. 2 ft. 8 in.; 
6 yd.1ft.9 in.;2 yd. 7 in., and 4 yd. 2 ft. 6 in., at 60c. 
per yd. How much did it cost? 


2. Put a wire fence, consisting of three wires, around 
a field that is one-half mile long and one-fourth mile wide. 
The posts are 16% feet apart. How much did it cost if 
wire is worth 10c. per rod (staples furnished) and posts, 
cost 10c. each? 


3. How much would it cost to carpet two rooms, 
17 ft. by 21 ft. and 15 ft. by 25 ft., carpet running length- 
wise and costing 70c. per yd.? 


4. Which would cost the most and how much? To 
tile a hall 8 ft. by 20 ft. with tiles 4 in. by 4 in. costing de. 
per dozen or to carpet it (lengthwise) with body Brussels 
three-fourths yard wide at $1.25 per yd.? 

5. I sell the oats in a bin 12 ft. by 6 ft. by 4 ft. at 22c. 
per bu. (Approx. 14 cu. ft.=1 bu.) and invested the pro- 
ceeds in molasses at 26c. per gallon. How many barrels 
and gallons did I buy? 


MANU-MENTAL COMPUTATION. 65 


6. Quinine costs $1.80 per oz. A pharmacist bought 
3 Ib. 8 oz. and sold it in 5 gr. doses at 10c. per dose. 
Find his gain. 

7. Bought 65 lb. licorice at 20c. per pound avoir- 
dupois and sold it at 28c. per pound apothecary weight. 
How much did I gain? 

8. Bought a nugget which weighed three pounds (Av. 
Wt.) for $230. One-half of it was gold. I sold this to the 
mint (Troy Wt.) at $20 per oz. Did I gain or lose? 

How much? 

9. A tank 8 ft. x 6 ft. x 12 ft. has 10 feet of water in it. 
How many gallons of water does it contain? (Approx. 
7% gallons=1 cu. ft.) 

10. A tank 5 ft. x 7 ft. x 9 ft. on the inside is filled 
with water. The tank weighs 350 pounds Av. How 
much weight is there on the foundation? (Approx. 1 
gallon weighs 83 pounds.) 

11. A field produces 46 bu. 2 pk. 6 qt. of corn per 
acre. I sold the crop from 37 acres at 35 cents per bu. 
How much did I receive? 

12. How many spoons can be made from 6 lb. 4 oz. 
10 pwt. of silver if one spoon weighs 2 oz. 5 pwt. 

13. I sell 6 T. 5 ewt. of coal at $6.50 per ton and buy 
flour at $4.75 per barrel. How many barrels do I get? 

14. I put a bin of wheat into sacks and find I have 42 
sacks of 2 bu. 1 pk. 2 qt. each. How much was in the 
bin? 

15. I carry 65 lb. 12 02. of: coal in=each bag. [I 
deliver 30 bags when coal is $6 per ton. How much 
should I receive? 

16. How many quinine pills of 3 gr. each can be made 
from 7 343209? 


66 MANU-MENTAL COMPUTATION. 
THE CALENDAR. 


96. To add any number of days to any date, add the 
number of days to the day of the month, divide by 30, the 
result will be months and days. 

Count forward the number of months, subtract 1 day 
for each “joint”? or “end” passed over and add one (for 
ieap year) or two days if February is passed. 


97. Illustration 32: 

Add 135 days to Jan. 24. 
24+135 =159 from Jan. Ist. 
159 +30=5 mo.+9 days. 
Jan.+5 mo.9 days =June9. 

As one end and two joints are passed over in the 5 
months, 3 must be subtracted from the 9 days=6 days. 
As February is one of the months passed, add 2 days= 
June 8. Therefore June 8 is 135 days later than Jan. 24. 


98. Illustration 33: 
Add 74 days to Feb. 6, 1904. 
Feb. 6+ 74 =Feb. 80th. 
80 +30 =2 mo.+ 20 days=Apr. 20. 
Feb. has 29 days, so subtract 1=Apr. 19. Mar. is on 
a joint, so add 1 and the result is Apr. 20. 


99, Illustration 34: 

Oct. 16+138 days. 

Oct. 16+138 =154 days. 

154+30=5 mo. 4 da. =Mar. 4. 

3 joints and ends are passed over, so subtract 3 
days from Mar. 4=Mar. 1. 

Add 2 days for Feb. =Mar. 3. 


100. To find the number of days between two dates, 
count the months on the fingers, multiply by 30, add 1 day 
for each joint or end passed over and subtract 1 or 2 for 
Feb., if it is one of the months completed. This gives the 


lel one, 
CALENDAR... 


68 MANU-MENTAL COMPUTATION. 


number of days from the first date to the same day of the 
last month. If the given day of the last month is before 
this day, subtract their difference; if it is after, add their 
difference and the result will be the number of days be- 
tween the two given dates. 


101. Illustration 35: 
Find the number of days from March 26 to July 15. 
Mar. to July =4 mo. 
4X30 =120. 
As 2 joints are passed over add 2. 
120+2=122. This is the time to July 26. 
26-15 =141, 
122-11=111. Number of days between March 26 
and July 15. . 


102. Llustration 36: 
Find the number of days from Jan. 18, 1904, to May. 
30, 1904. 
4 mos. =120. 
1 joint and 1 end =2. 
120+ 2=122. 
Feb. has 29 days, so subtract 1 day. 122—1=121. 
30—18 =12, 
121+12=133 days between Jan. 18 and May 30. 
Nore 24.—To find the number of months and days 
between two dates count them on the fingers. To find 
the date that is any number of months and days: after a 
given date count forward that many months and add the 
days. The student will learn this method in a very short 
time. 
103. Problems on the Calendar. 
Find the number of days between 
1. Jan. 7 and Apr. 25. 
2. Mar. 27 and Oct. 3. 
3. July 16 and Dee. 28. 
4. Nov. 23 and May 8 (next year). 
5. Dec. 26,1903, and July 4, 1904. 


MANU-MENTAL COMPUTATION. 69 


Find the number of years, months, and days, between 


6. Jan. 13 and Oct. 23, 1900. 

7. July 26, 1902, and Dec. 4, 1904. 

8. July 30, 1902, and Feb. 29, 1904. 

9. Nov. 28, 1889, and Mar. 21, 1906. 
10. July 4, 1776, and June 26, 1904. 


Find date 


11. 35 days after Jan. 15. 

12. 178 days after Nov. 12. 

13. 23 days after Dec. 31. 

14. 137 days after Feb. 6. 

15. 267 days after July 12. 

16. 3 mo. 10 da. after Feb. 7, 1892. 

17. 5 mo. 29 da. after June 30, 1896. 

18. 2 yr. 5 mo. 15 da. after Sept. 23, 1901. 

19. 7 yr. 19 da. after June 1, 1905. 

20. 16 yr. 7 mo. 24 da. after Dec. 24, 1876. 

21. How old are you in years, months, days? 

22. I was born Jan. 10, 1870. How old was I July 4, 
1904? . 

23. Two notes were given Oct. 1, 1903, one for 6 mo. 
10 da. and the other for 190 da. Find when each is due. 

24. Two notes were given Jan. 15, 1902, one for 2 mo. 
5 da.; the other for 75 da. Find when each is due. 

25. How many days since July 24, last? 

26. A note was given Jan. 29, 1903, and paid July 3, 
1904. How many days did it run? 

27. A note was given June 21, 1903, and paid Mar. 2, 
1904. How many days did it run? 


70 MANU-MENTAL COMPUTATION, 


INTEREST AND DISCOUNT. 


104. Most practical problems in interest can be worked 
by the Manu-Mental Process. It is especially applicable 
to the Bankers’ Method; 7. ¢., shift the decimal point two. 
places to the left for the interest for 60 days at 6%, or 
three places for 6 days at 6%. For other times and 
rates use proportion or aliquot parts. 

Bank discount is the same as simple interest. True 
discount is the amount which deducted would leave a sum 
which if placed on interest for the given rate and time 
would produce the amount of the note or bill discounted. 


Note 25.—The point shifted two places to the left 
gives the interest for 


180 ae at 2% 45 daysat8% 
120 3 40 3 9 
90 i 4 36 ea O 
72 : 5 30 ent ce 
60 < 6 
105. Problemsin Interest and Discount: 
1. $200 at 6% 3 yr. 
2. $130 at 6% 14 yr. 
3. $420 at 607, 1 yr.9 mo. 
4. $750 at 6% 1 yr. 4 mo. 12 da. 
5. $300 at 8% 3 yr. 
6. $240 at 8% 24 yr. 
7. $635 at 8% 1 yr. 3 mo. 
8. $145 at 4% 2 yr. 4 mo. 18 da. 
9. $60 at 5% 4 mo. 
10. $25.30 at 6% 18 da. 


—_ 
— 


. $15.62 at 4% 1 mo. 20 da. 

. $700 at 74% 2 yr. 3 mo. 12 da. 
. $30.45 at 8% 15 da. 

. $163 at 9% 3 mo. 9 da. 

. $12.60 at 8% 10 da. 

. $2465.20 at 4% 30 da. 


a or a or 
Ook WLW LO 


MANU-MENTAL COMPUTATION. 71 


17. $5400 at 8% 30 da. 

18. $3870 at 9% 90 da. 

19. $4065 at 3% 125 da. 

20. $1777.40 at 4% 6 mo. 10 da. 

21. Find the bank discount on $425 at 5% for 4 mo. 
20 da. 

22. Find the true discount on $4343 at 6% for two 
months. 

23. Find the true discount on $591.23 at 8% for 90 da. 

24. A note for $67 for 3 months at 8% was paid at 
maturity. How much was paid? 

25. A note given Mar. 17, 1903, wes paid Jan. 15, 
1904. How much was paid if the rate was 9%? 

26. A note for $635 was due July 2, 1904. It was 
discounted at the bank June 12th at 5%. How much 
was received? 

27. Note for $420 at 8% dated April 20, 1904, is due 
90 da. after date but is not paid until ten days after due. 
Find amount paid. 

28. Note for $76.40 at 4% dated June 3 is paid July 
6. Find amount. 

29. Note for $55.50 at 6% dated Aug. 26 to run 210 
da. is paid 10 da. before due. Find amount paid and date 
When note was due. 

30. Note for $135 for 3 mo. was discounted at the 
bank 20 da. before due. How much did the owner 
receive? 

31. Note for $60 at 7% dated Oct. 12, due Dec. 27, 
was discounted at the bank Nov. 14 at 6%. How much 
did the owner receive? 

32. Note for $1250 at 5% dated Dec. 4, 1903, for 90 da. 
was discounted Feb. 21, 1904, at 6%. Find amount paid. 

33. A man owes me $125, due in 6 mo. How much 
will settle the bill now at true discount, if money is worth 


8%? 


72 MANU-MENTAL COMPUTATION. 


FACTORS. 


106. Most numbers below 250000 can be factored by the 
Manu-Mental process, by dividing by any divisor and 
recording on the fingers. 

A divisor or factor which is more than ten and less than 
twenty-one can be recorded on one finger by placing the 
pointer against the right side of the record finger, instead 
of directly on it. 

A divisor or factor which is more than twenty and less 
than thirty-one may be recorded on one finger by placing 
the pointeragainst the left side of therecord finger instead 
of directly on it. (Fig. 21.) 


Nore 26.—Have the student learn all the prime factors 
below 30. He would find it convenient to know all the 
prime factors below 100. 


107. Llustration 37. 
Factor 990. 
Record on fingers. 
Divide by 2 (recording on thumh) = 495. 
Divide 495 by 3 (recording on first finger) =165. 
Divide 165 by 3 (recording on 2d finger) = 
Divide 55 by 5 (recording on 3d finger) =11 (recording 
on 4th finger). 
Then the factors are 2,3, 3,5, 11. 


108. Illustration 38. 
Factor 60697. 

Record on fingers. 

Divide by 7 (record on thumb) =8671. 

Divide by 13 (record on Ist finger) =667. 

Divide by 23 (record on 2d finger) =29 (record on the 
3d finger). 

Then factors are 7, 13, 23, 29. 


74 MANU-MENTAL COMPUTATION, 


109. Problems in Factoring: 


Find the factor of Find the prime factor of 
li 75 ily 98 
2. 46 2. 45 
3. 64 3. 64 
4, M75 4, 81 
5. 238 5. 125 
6. 346 6. 420 
77 21260 (eS hs 
8. 2796 8. 844 
9. 146146 + 9+ 21504 

10. 278278 10. 124416 


110. A factor is a divisor. A factor of two or more 
numbers is called a common factor or common divisor. 
The largest common factor of two or more numbers is 
called the greatest common divisor or highest common 


factor. 


111. To find the G. C. D. of two numbers, divide the 
smaller into the greater. If there is no remainder the 
smaller isthe G. C.D. If there isa remainder, divide this 
remainder into the previous divisor. Continue dividing 
the last divisor by the remainder until there is no re- 
mainder. The last divisor is the G. C. D. 

To find the G. C. D. of more than two numbers, find the 
G. C. D. of two; find the G. C. D.*of this G.C. D. and 
the next number; continue this until the last number is 
used. The result will be the G. C. D. of all. TheG. C.D. 
may be found by factoring the first number and discarding 
all the factors not found in the others. The product of 
these factors is the G.C, D. 


112. Find the Greatest CommontDacecnts 


1. 62°91. 6. 423, 2313. 
3°65, 143) 7. 18584, 24610. 

3. 66,165. 8. 1365, 2093, 2205, 707, 13013. 
4, 192, 460. 9. 707, 4949, 16463. 


5. 36,63, 162,288. 10. 86,344, 47343, 51686. 


Or 


MANU-MENTAL COMPUTATION, © 4 
MULTIPLES. 


113. A multiple of any number may be found by multi- 
plying it by any whole number. 


A multiple of any two numbers (called common multi- 
ple) may be found, Ist, by multiplying them together; 
2d, by dividing both by any conimon factor and multi- 
plying these quotients and this common factor together; 
3d, by factoring the numbers to the prime factors, select- 
ing each prime factor as many times as it is found in any 
one of the numbers, and multiplying these prime factors 
together (this is the least common multiple); 4th, by mul- 
tiplying the L.C. M. by any whole number. 

To find the L. C. M. of three or more numbers, find the 
" L. C. M. of two, then find the L. C. M. of this multiple and 
the next number. Continue this process until all the num- 
bers are used. The final L. C. M. is the L. C. M. of all the 
numbers. 


{14. Find the Least Common Multiple of 


4; 9,1 2: 6. 69,161, 153. 
2. 4, 7,12. 7. 15, 30,75, 105. 
OstOy 1D, a0 4 8. 45, 48, 80, 135. 
4. 8,12, 40, 60. Oo 20cu nL ola 
5. 17,85, 153, 1836. . 10. 5226, 7839. 


11. What is the shortest distance that can be meas- 
ured exactly by a 2, 5, 6, or 8 foot rod? 

12. I must divide the money I have in my pocket 
among 5, 6, or 7 boys and give each the same number of 
dollars. What is the smallest sum I can have in my 
pocket? 

13. A, B and C can walk around the race course in 
9, 12, and 15 minutes, respectively. If they start together 
and each does his best, how long will it be before they are 
together again at the starting point? 


76 ' MANU-MENTAL COMPUTATION. 


ALIQUOT PARTS. 


115. Aliquot parts are simply problems for multiplica- 
tion and division, and are easily performed by the 
Manu-Mental process by applying the instructions given 
in any good arithmetic. 

116. Problems in Aliquot Parts: 

Find the cost of 7 
. 160 yd. cloth at 75c. per yd. 

. 75 yd. cloth at 334c. per yd. 

. 1250 lb. prunes at 64c. per lb. 

. 2400 bu. corn at 374c. per bu. 

. 750 bu. oats at 333. per bu. 

718 lb. wheat at $1.25 per bu. 

. 14250 lb. coal at $6.66% per ton. 
. 22550 Ib. hay at 124c. per ewt. 

. 63 doz. cans corn at 624c: per doz. 

10. Sold 22 hogs at $9.09$ each and bought corn 
at 20c. per bu. How much corn did I get? 

11. Worked 16 days at $2.25 and 24 days at $2.334 
per day. Spent $12 and invested the remainder in ducks 
at $8 per dozen. How many ducks did I get? 

12. Sold 40 head of cattle averaging 1200 lb. at 64e. 
per Ib. Invested the poceeds in wheat at 75c. per bu. 
How many bu. did I get? 

13. Bought a farm of 320 acres at $14.284 per acre. 
Sold one-fourth of it at $15 per acre and the remainder at 
$16.662 per acre. How much did I gain? 

‘14. A wagon that weighs 1250 pounds weighs 5090 
pounds when loaded with coal. What is the load worth if 
coal is 163c. per bu.? 

15. Bought 14 yd. of platinum wire at 24c. per in., 
used 3 ft. 2 in. of it and sold the remainder at 34c. per in. 
How much did the part I used cost me? 


— 


CONAaAwWb 


MANU-MENTAL COMPUTATION. : i 
METRIC LENGTH MEASURE. 


117. In measuring long distances the meter is the 
smallest measure generally used: in measuring short dis- 
tances the meter is the largest measure generally used. 
As most practical problems deal with the larger or the 
smaller only, they can be worked on the hand by using the 
little finger for mm. or M. (Fig. 22.) 

If not more than five of the orders are used they may be 
shifted on the hand, placing meters or any other denomi- 
nation on any convenient finger; if the problem requires 
KM. to dm., M. may be placed on the third finger, ete. 

118. The meter, which is the standard of length, is 
39.3685 inches. For approximate measures consider it 3.3 
feet, or 1.1 yards. 

119. Problems in addition, subtraction, multiplication, 
division, reduction ascending, reduction descending or 
reduction to or from long measure are easily performed by 
applying the instruction previously given in this book. 


METRIC SQUARE MEASURE AND 
METRIC CUBIC MEASURE. 

120. While 100 or 1000 cannot be recorded on one finger, 
the student will find, if he has been thoroughly drilled on 
all preceding work, that he can remember the numbers 
of each denomination with sufficient accuracy to enable 
him to solve most practical problems in surface and volume 
measure, 

121. The standard for surface measure is the square 
dekameter, which is called Are and equals 3.954 sq. rd. 
The square meter is called a Centare and equals 1.196 
sq. yd. The ratio of surface measure is 100. 

122. The standard for solid measure is the cubic meter, 
called Stere, and equals 1.3078 cu. yd. The ratio of 
solid measure is 1,000, 


78 MANU-MENTAL COMPUTATION. 


123. The Liter is the un‘t of capacity. It equals 1 
cu. dm. 1.0567 liquid qt. or .908 dry quarts. 


124. The standard for weight measure is the Gram, 
which equals 15.432 grains troy. (Av. lb. =7000 gr. Troy 
and Apoth. lb. =5760 gr.) 


125. Problems in the Metric Systems: 


1. Reduce 27 m.to cm. 

2 4 Hg. to dg. 

3 c 6 DI. to dl. 

4, (eA Km, osities mato um: 

5. i 9 dg. 5 eg. 2 mg. to g. 

6 st 21.6 cl. 4 ml. to Kl. 

‘p ‘ 12 sq. Hm. 65 sq. Dm. 10 sq. m. 86 sq. dm. 

to sq. m. 
8. Reduce 4 DI. to bu. 
Oe ee Oto ral; 


10. i 5 Ares to sq. rd. 

Tu r 4 Mg. to lb. Troy Wt. 

Le 3 7 Q. to lb. Avoirdupois Wt. 
13. ¥ 12 bu. to liters. 


14. a 5 gal. 1 qt. (liquid) to liters. 

15. a 25 sq. yd. to sq. m. 

16, y 28 sq.m. to sq. ft. 

17. Bought 200 bu. oats at 22c. and sold them at 
$1.50 per HI. after paying $16 shipping charges. How 
much did I gain or lose? 

18. Bought 20 Steres of wood at $1.50 per Stere and 
sold it at $6.50 per cord. Find gain or loss. 

19. Bought 25 Steres of wood at 8 fr. per Stere and 
sold it at $7 per cord. Find gain or loss. 

20. Bought 15 cords of wood at $8 per cord and sold it 
at 7 fr. per Stere. Find gain or loss. 


FIG. 22. I 


16. 
VIETRIC LENGTH MEASURE. METRIC bier y meGwee 


FIG. . FIG. 
METRIC WEIGHT MEASURE. METRIC S$ Pinte pe. URE, 


80 MANU-MENTAL COMPUTATION. 
FRACTIONS. 


126. Fractions whose numerators and denominators are 
less than one hundred may be recorded on the fingers by 
using the first and second fingers for the numerator and 
the third and fourth fingers for the denominator. When 
this is done it is well to spread the third and fourth fingers 
as in figure 7. 


127. Reduction of fractions is performed by recording 
the numerator and denominator as above and multiplying 
or dividing both. 


128. To add fractions, find the common denominator 
(L. C. M., page 75,) remember it, but do not record it on 
the fingers. Reduce each fraction separately to a fraction 
having that denominator and add the numerators, record- 
ing the operation and result on the fingers, 


129. Subtraction of fractions may be performed by 
reducing to a common denominator, recording the numer- 
ators and subtracting as if they were whole numbers. The 
result is the numerator only. 


130. In multiplication of fractions the memory must 
retain the denominators while the numerators are multi- 
plied, and vice versa. 


131. Fractions are divided by inverting the divisor and 
multiplying the fractions. (Not theoretically correct.) 


Note 27.—The work in fractions must be largely mental, 
as the memory must retain all that cannot be recorded on 
the hand. In computing with whole numbers and frac- 
tions it will often be necessary to separate them and per- 
form the operations separately, then combine the results. 


“MANU-MENTAL COMPUTATION. Sl 


132. Problems in Fractions: 
1. Reduce: 2 to 12ths. 
“ 4 to 60ths. 
: + to 68ths. 
a 2¢ to 3rds. 
= 46 to 20ths. 
23 to lowest terms. 
zz to lowest terms. 


~ 


go to cowest terms. 

§4 to lowest terms. 
. 2 ae B 

Add: st+3t+e+2. 


pet 
rH SONA wo) 


6 16 2 13 
12. atte t 43 

e 3 2 9 
13% Pes aes 
14, Subtract: 37-4. 
15. i $4 —§, 

ce 13 iE 
16, Sl Geena 

7 ~s 2 6 14 
17. Multiply: 2x$ x44. 

‘sé 7 9g Vs 
13; aX XaR- 
19: = EXEXdE. 
20. Divide: 6-+2. 

““c a Se 
7A W, 2-7, 
92 «“ | eed 
oo ed 8 . Q. 

‘ eg ER: 
aA BE Boe do Ss 
aa ESS Wa 2 
25. If 2 of a suit of clothes is worth $74, how many 


days work at $14 will pay for the suit? 

26. Corn is worth $3 per bu. I sell a load of 25 bu. 
and buy oats at $4 per.bu. How many bu. of oats do I 
get? 

27. I bought ? of a farm and sell ? of my share for 
4 of the cost of the whole farm. What % will I gain if 
I sell the remainder of my purchase at the same rate? 

28. B owns 4 of a tract of land. C owns 3 as much 
as B. D owns } as much as both B and C. E owns the 
remainder, What part of the whole does he own? 


82 MANU-MENTAL COMPUTATION. 


29. Spent $ of my money for board, 4 of the remainder 
for clothes, and 4 of the remainder, which is $10, for car- 
fare. How much had [I at first? 


30. I increased my money by + of itself at one time 
. and 3 of itself at another. After spending 2 of what I 
then had, I increased the sum I then had by # of itself 
and found I had $164. How much had I at first? 


BUSINESS COMPUTATIONS. 


133. The solution of problems in Percentage, Profit 
and Loss, Trade Discount, Commission, Equation of Ac- 
counts, Custom House Duties, Taxes, Insurance, Stocks 
and Bonds, Exchange, Partnership, ete., consitsts of two 
parts; viz., analysis and computation. The analysis 
must be mental primarily. This book gives sufficient 
instruction to cover most, if not all, computation needed 
in practical problems; so that the pencil or chalk need not 
be used to any great extent. 


Notre 28. The following problems should be worked 
without using pencil or chalk. After the student has been 
thoroughly drilled on the preceding principles and prob- 
lems he can work most of these mentally, The remainder 
of the problems should be worked manu-mentally. There 
_are but very few problems in school or business arith- 
metics that cannot be solved by the manu-mental method. 


MISCELLANEOUS PROBLEMS. 
1. Paid A $2,467, B $5,478, C $986, D $1,545, and E 
$2239. How much did I pay all? 


2. Bought beef $134.20, lard $35.65, sugar $86.32, 
syrup $68.95, rice. $26.10, pork $247.80, flour $306 and 
apples $82.05. Find amount of bill. 


MANU-MENTAL COMPUTATION. §3 


3. Add 2468, 3507, 9640, 428, 1007 and subtract 862, 
587, 5786, 2050 from the sum. 


4. Bought for my country store as follows: gfoceries 
$2,647, boots and shoes $345, hats and caps $146, clothing 
$1,431. At the end of the year I find I have sold as 
follows: groceries $2,132, boots and shoes $357, hats and 
caps $105, clothing $1,500 and have an inventory of $1638. 
How much did I gain or lose? 


5. A book contains 345 pages. How many pages in 
8 volumes? 


6. A has $2346 and B has 23 times as much. How 
much has B? 


7. C has 514 cattle which he values at $78 a head. 
D offered him $1235 less than he asks. What did D offer 
for the herd? How much per head? 


8. A pipe flows 41 gallons per hour. How long will it 
take to fill a tank that holds 14432 gallons? 


9. The school tax was $76229. $965 were spent for 
repairs and the remainder was divided among 94 teachers. 
How much did each teacher get? 


10. I spent $235 in one month, At that rate how 
much would I spend in 74 months? In 124 months? 


11. A company of 24 men have provisions enough 
for 88 days, and another company of 52 men have pro- 
visions enough for 247 days. If the companies are 
united and the provisions combined, how long will the 
provisions last? 


12. A drover has twice as many hogs as sheep and 
three times as many sheep as cattle. If the hogs cost $7, 
the sheep $5, and the cattle $24 each, how many of each 
did he buy for $5913? 


84 MANU-MENTAL COMPUTATION. 


13. If 28 men can do a peice of work in 46 days, how 
many men must be employed to complete the work in 8 
days? 


14. If 15 men complete a piece of work in 16 days, 
how many men must be added to complete it in 12 days? 


15. I walk 324 miles in 72 hours. How far can I 
walk during the month of February, 1904, if I walked 8 | 
hours each day? 


16. An ocean steamer travels 21 miles an hour when 
there is no wind. In crossing to Europe she faces a wind 
which retards her 4 miles per hour and makes the trip in 
214 hours. On the return trip the wind increases her 
speed 3 miles per hour. How long does it take her to 
make the return trip? 


17. 20 men working 123 days of 8 hours each do a 
piece of work. How many men will it require to complete 
the work in 205 days working 6 hours per day? 


18. A country store keeper exchanged 4 loads, of 42 
bushels each, of wheat worth 55 cents per bushel for 5 
bolts of cloth, of 104 yards each. Find the cost per yard. 


19. I rode a motor bicycle 23 days of 15 hours each 
at the rate of 14 miles per hour. I had the following 
expenses for wheel: gasoline $8, chain $4.60, 4 spokes $1, 
saddle spring $1.50, oil 75 cents, tire tape 25 cents. How 
much per mile did my ride cost? 


20. If the wheat in a bin 8 ft. high, 12 ft. wide and 24 
ft. long is worth $1152, what is the wheat in a bin 15 ft. 
high, 20 ft. wide and 30 ft. long worth? 


21. A, B and C rent a house for $606 per year. A 
occupies 8 rooms, B 6 rooms and C 11 rooms. Hew much 
per month should each pay? 


MANU-MENTAL COMPUTATION. 85 


2. E and F own 1948 sheep. Two times the number 
E has plus 73 equals the number F has. How many have 
each? | 


23. Built astone wall 120 ft. long, 8 ft. high and 14 ft. 
thick at $3 per perch. Find cost. 


24. How many bricks in a wall around a lot 30 ft. by 
50 ft. (on the inside) if the wall’is 4 ft. high and 1 ft. thick 
and contains 26 bricks per cu. ft.? 


25. A house 60 ft. long, 35 ft. wide and 20 ft. high 
(outside measurement) has walls 18 in. thick. If deduc- 
tions for doors and windows will build the gables, what 
will the brick cost at $8 per thousand? 


26. I have an income of $2000 per year. I spend 25 
& for board and room, 10 % for clothing, 124 % for books, 
20 % for traveling, etc. How much do I spend for each 
and how much do I save? 


27. I own 60 % of a farm and sell 35 % of my share 
for $3405. What is the farm worth? 


28. I gave John 30 % of my bank account, paid 20 % 
of the remainder for insurance, spent + of the remainder 
for clothes and had $1200 left. Find how much I had at 
first and how much I spent for each item. 


29. Give 4 of my property to charity, 5% of the re- 
mainder to my wife, 75 % of the remainder to my son, 
50 °% of the remainder to my daughter and the remainder, 
$1000, to my brother. Find value of property and how 
much each got. 


30. 20 half chests of Oolong tea containing 75 pounds 
each at 35 cents per pound were exchanged for coffee at 
22 cents per pound. How many sacks of coffee were 
given if a sack weighs 125 pounds? 


86 MANU-MENTAL COMPUTATION. 


31. A wheel, which has 26 cogs, turns a wheel, of ten 
cogs, that is on the same rod with a wheel of 32 cogs 
which turns a wheel of 6 cogs. If the 26-cog wheel 
makes 15 revolutions, how many revolutions will the 
6-cog wheel make? 


32. Bought a farm of 230 acres at $22.50 per acre. 
Sold it at a gain of 50 %. After taking out $262.50 for 
expenses, I invested the remainder in land at $15 per acre. 
and sold it at a gain of $2 per acre. How much did I gain 
by the total transaction? 


33. Bought 28 tons of R.R. rails in Pittsburg, Pa., 
and sold them at $32 per ton in Sioux City, Ia., after 
paying $135 for freight. How much did I gain or lose? 


34. A produce dealer paid $360 for pears, $150 for 
peaches, and $130 for apples. He sold the pears at 25 % 
profit, the peaches at 334 % loss, and the apples for 90 % 
of the cost. What did he lose or gain? 


35. A dealer bought goods and paid 1632 % of the cost 
for freight. He sold one-half of the goods at 28 4 % gain 
and the other half at 57+ % gain and gained $600 by the 
transaction. Find cost of goods, and amount received for 
each sale. 


36. I bought four horses for $150, $225, $400 and 
$650 respectively. Sold the first at a gain of 50 %, the 
second at a loss of 114 %, the third at a gain of 64 %, 
and the fourth at a gain of 7%; %. Find total loss or gain. 


37. I paid $540 for hardware. Sold $150 worth to 
one man, and $210 worth to another and find I have $240 
worth remaining. What per cent, and how much profit 
did I make? 


38. I bought two horses at $125 each; sold one at a 
gain of 20 %, and another at a loss of 20 %. How much 
did I gain or lose on both? 


MANU-MENTAL COMPUTATION. 87 


39. I sold two horses at $125 each; on one I gained 
25 %, on the other I lost 25 %. How much did I gain or 
lose on both? 


40. I sold a house at 10 % gain; with this money I 
bought another and sold it at 20 % gain. My total gain 
was $550. What did each house cost? 


41. Bought a farm of 960 acres in Missouri at $12.50 
per acre; built a house which cost $2420; a barn which cost 
$3600 and paid $1375 for fencing. I sold 120 acres at $15 
per acre, 240 acres at $25 per acre and the remainder 
(@neluding barn and house) at $110 per acre. Find the net 
gain or loss. 

42. Bought a note for 124 % less than its face and 
discounted it at a bank at 6% for 60 days. Find my gain 
or loss. 

43. Sold $450 worth of merchandise at 334 % and 
20 % off. Find net bill to render. 

44. Sold $3500 worth of merchandise at a discount of 


50 %, 20 % and 10 %. Find net per cent discount and 
net amount of bill. 


45. A merchant buys clothing at 30 % and 20 % 
discount from marked value’and sell it at 20 % above 
marked price. How much does he gain? 

46. A merchant buys goods at 30 % and 20 % off the 
marked price and sells it at a discount of 10% and 20% off 
the marked price. Find his net Joss or gain. 

47. A discount of 50 %, 30 %, 20 %,10 % and 5 %G is 
equal to what single discount? ; 

48. Bought $3400 worth of merchandise at 24 % 
commission. Find my commission. 

49. Sold 500 bu. of wheat at 80 cents a bu. Paid 


commission 3 %, insurance $25, freight $130. How much 
did I receive? 


88 MANU-MENTAL COMPUTATION. 


50. Sent my agent $1200 to invest after taking out 
his commission of 3 %. How much did he invest for me? 


51. Shipped a carload of 30 cattle to Chicago; they 
sold for $32.50 each. The commission merchant took out 
4% commission and invested the remainder in wheat after 
deducting his commission of 2% for buying. What was 
his total commission and how much wheat did he buy at 
75 cents a bushel? 


52. An agent sold oats on 5 % commission and in- 
vested the proceeds in wheat at 75 cents per bushel on a 
commission of 53 %. For how much did he sell the oats 
if his total commission was $300? 


53. Find the interest on $420 at 6 % for 25 days. 
54. Find the interest on $950 for 5 months at 9. %. 
55. Find the interest on $2560 for 70 days at 8 %. 


56. A note for $420 dated June 3, 1903, was paid Oct. 
5, 1903, with 6 %. Find amount paid. 


57. A note for $3240 dated Feb. 3, 1894, for 60 days 
without interest was paid Apr. 25, with 6 % interest after 
due. Find amount paid. 


58. A note for $785.10 dated Sept. 23 for 45 days 
without interest was paid Nov. 17 with 8 % interest from 
due. Find amount paid. 


59. A note for $400 dated Mch. 4,.1904, was paid 
May 1 with 9% interest from Feb. 20. Find amount of 
payment. 


60. A note for $300 is dated Jan. 1, 1904, at 6%. The 
following payments were made: Feb. 10, 1904, $22; 
Mch. 11, 1904, $1; Apr. 25, 1904, $82.50. What was due 
June 7, 1904, by the United States rule for partial pay- 
ments? 


MANU-MENTAL COMPUTATION. 89 


61. A note for $465.10 at 8 % interest is dated June 
12, 1902. The following payments are endorsed on it: 
Aug. 20, $51.84; Oct. 12,$5; Nov. 4,$25. What is due 
Dec. 12, 1902? 


62. A note for $1430 at 8% interest, dated Apr. 23, 
1902, has the following endorsements: June 10, 1902, $50; 
Aug. 28, 1902, $10.50; Nov. 4, 1902, $100; Dec. 20, 1902, 

$10; Mch. 1, 1903, $15.10; May 25, 1903; $50; June 10, 
1908, $35.45; Sept. 30, 1903, $325.30. What was due 
Dec. 3, 1903? 


63. A man invested $12500 in business. At the end 
of 24 years he drew out $17000, which represent the invest- 
ment and gain. If his services were worth $1200 per 
year, what per cent did he receive on his investment? 


64. A building costs $125000; receipts from rents, etc., 
produce $34500, lighting and heating costs $8400, taxes 
$5000, insurance $2400, repairs $3500, janitor service, etc., 
$2000. If the agent charges 3% for collecting, what per 
cent does the investment net me? 


65. I loaned $3500 at 8 % compound interest for 3 
vears 4 months. How much should I receive? 


66. I paid $800 freight on a carload of goods received 
from New Orleans. The insurance and drayage cost me 
$130. I sold the goods for $3600.85 and thereby gained 
164 % on the total cost. Find what the goods cost in 
New Orleans. 


67. I invested $52000 in business. The first year I 
gained 6 %, the second year 15 %, the third year 45 %, the 
fourth year I lost 10%. If the gain was added to the 
capital each time and the per cent of gain or loss taken on 
the new capital, what was the business worth at the end of 
the fourth year and what was my net per cent gain on the 
original capital? 


90 MANU-MENTAL COMPUTATION, 


68. A man loaned me $500 for four months at one 
time and $800 for nine months at another time without 
interest. For how long should I lend him $1000 to 
balance the accommodation. ? 


69. I owe a man $400, payable today, and $800, pay- 
able in 24 days. By his consent I decide to pay it all in 
one payment. When should the settlement be made? 


70. March 1, I buy $800 worth of goods on 30 days © 
time, $500 worth of goods on 40 days time and $1000 
worth of goods at 60 days. When should I make settle- 
ment if I paid the three bills at one time? 


71. On April 10, I purchase $450 worth of goods at 90 
days, Apr. 15, $800 at 30 days, Apr. 28, $200 at 10 days, 
May 10, $800 at 60 days, June 4, $850 at 30 days. What 
should be the date of settlement, if the bills were all paid 
at once? 


_ 72. I owe $500 which should be paid on Apr. 23, I 
pay $200 Apr. 10. When should the remainder be paid? 


73. Find the specific duty on 850 pounds of coffee at 
4c., 3500 pounds tea at 10c., 500 pounds rice at $e. 


74. Find the ad valorem duty on 450 yd. of cloth at 
Sc., 520 vd. of carpet at 65c., 345 bottles perfume at 55c. 
and five cases of drugs at $22. If the ad valorem duty for 
the drugs and perfume is 65% and the cloth and carpets 
10%. 

75. Find the duty on 1550 yd. of silk invoiced at 


$1.10 per yd. with 15c. a yard specific, and 2 % ad valorem 
duty. 


76. I imported 500 pounds woolen goods invoiced at 
£620. The duty is 224 cents per pound and 30% ad 
valorem. If I sell the goods at a gain of 20 % on the 
total cost, how much do I receive? 


MANU-MENTAL COMPUTATION. 9] 


77. I imported 320 dozen German, brass lined pocket 
knives, valued at 8 marks per dozen, with a specific duty 
of 25 cents per dozen and 20 % ad valorem. If I sell them 
at a gain of 40 %, how much do I receive? 


78. Imported 3 doz. (100 Ib. each) Swiss cheese 
invoiced at 2 centimes per milligram (Approx. 2.2 lb., Av.) 
with 25 % ad valorem and $1.20 per cwt. specific duty. 
For what must I sell it to gain 20% on the total cost? 


79. I owned $4,500 real estate and $2,750 personal 
property, the rate of taxation 3.5 mills on the dollar. How 
much do I pay? 


80. The property is a village js valued at $240,000. 
There are 130 polls at $1.25 each. The rate of taxation 
is 2.6 mills on the dollar; I own $5,000 worth of property | 
and pay one poll tax. How much tax do I pay, and how 
much does the village raise? 


81. The taxable property in a town is valued at 
$650,000. There are 535 polls at $1.25. The total tax 
raised is $15,293.75. What is the rate of property taxa- 
tion? 


82. The property in a town is assessed at $12,500,000, 
the rate of taxation is 2? cents, there are 450 polls at 
$1.50. What is the total tax, and what does A pay, if he 
owns $10,250 worth of property and pays one poll? 


83. Insured my property for $12,000 at 3%. How 
much premium do I pay? 


84. My property is valued at $7,000. I insure one- 
half of it in one company at 4 % and one-third of it in 
another at 3%. If the premium is paid in advance and 
the property is entirely destroyed, what is my loss? 


92 MANU-MENTAL COMPUTATION. 


85. A company insures a block for $250,000 at 75 
cents per hundred, but thinking the risk too great, they 
reinsure one-fourth of the block in another company at 
8 mills and two-fifths of the block in a third company at 
84 mills. How much risk does the original company 
carry and how much does it get for carrying this risk? 


86. My factory is insured with one company for 
$4,000 and with another for $9,000. $15,000 damage is 
done by fire. How much do I lose and how much does 


each company lose, if my rate of insurance was 75 cents 
per $100? 


87. The Baltic is valued at $6,500,000 and is insured . 
for $1,500,000 on a pro rata policy at 24%. A fire does 
$325,000 damage; what is the premium and how much 
damage does the insurance company pay? 


88. I bought 50 shares of stock at 95, 20 shares at 104 
and 75 shares at 106. How much did I pay for all? 


_ 89. I bought 265 shares of American Steel stock at 
110 and paid 4 % brokerage. How much did it cost me? 


90. I purchased 50 shares C. B. & Q. R.R. stock at 
113% and 50 shares N. Y. Central at 119 through my 
broker. I sold the C. B. & Q. R.R. stock at 115 and the 
N. Y. Central stock at 120. If my broker charged 4% 
for buying and 2 % for selling, how much do I gain on the 
transaction? 


91. Traded 100 shares of D. L. & W. R.R. stock at 
965 for 225 shares of L. & N. R.R. stock What does one 
share of L. & N. stock cost me? 


92. Paid my broker $250 for selling stock ; brokerage 
4%. How much stock did he sell at par? 


MANU-MENTAL COMPUTATION. 93 


93. I paid my broker $4,342.50 to cover purchase and 
brokerage of 45 shares of Atchison, Topeka and Santa Fe 
R.R. stock. What does stock sell at? 


_ 94, The Gardner Governor Works has a capital of 
$300,000. The gross earnings are $69,520, the gross ex- 
penses are $27,260. $15,000 worth of bonds are paid off, 
what even per cent dividend can be declared and how 
much would be left for the surplus fund? 


95. We paid $8,400 on outstanding bonds, $17,000 
expenses, declared a dividend of 6 % and placed $7,600 in 
the surplus fund. What is the capitalization of the com- 
pany? . 


96. A, B, and C formed a partnership. A invested 
$800, B invested $1,000 ,C invested $1,300. They gained 
$1,240. How much should each receive? 


97. F and G formed a partnership. F invested 
$5,850 for 10 months, G invested $2,960 for 15 months. 
They gained $5,145. How much should each have? 


98. H and J formed a partnership. H_ invested, 
$4,500, J invested $500 and devotes his time to the busi- 
ness. If his work is worth one-third of his investment 
how much of the $2,600 gain should each receive? If the 
partnership is dissolved at the end of the year, how much 
should each receive, including investments and profits? 


99. An estate has to be divided among the wife, two 
sons and three daughters. The wife is to have one-third 
of the property, each son is to have twice as much as each 
daughter. If the property is worth $31,500, how much 
did each receive? 


Q4 MANU-MENTAL COMPUTATION. 


100. X, Y and Z formed a partnership for one year; 
each to devote his entire time to the business. X invests 
$2,000 and withdraws $1,000 at the end of nine months; 
Y invests $3,000 and at the end of six months, $1,000 
more; Z invests $4,000 and withdraws $2,000 at the end 
of eight months. Owing to illness X was unable to attend 
to business for three months. If the labor of each man is 
considered as worth $2,000 per vear, how should a profit 
of $5,025 be divided? 


THE END. 


‘ANON 


3 0112 017102531 


